An urn contains n black balls and n white balls. Three balls are chosen from the urn at random and without replacement. What is the value of n if the probability is 1/12 that all three balls are white

a. The probability is 0.30. The advertisement seems to have a positive effect on customer purchases.

b. The market share is 0.20. The advertisement could potentially increase the company's market share

c. The conditional purchase probability of 0.333. The second advertisement seems to have had a slightly bigger effect on customer purchases.

a. We are asked to find the probability of an individual purchasing the product given that the individual recalls seeing the advertisement, i.e., P(B|S).

Using the formula for conditional probability:

P(B|S) = P(B∩S) / P(S)

Given:

P(B∩S) = 0.12

P(S) = 0.40

So, P(B|S)

= 0.12 / 0.40

= 0.30

Seeing the advertisement increases the probability that an individual will purchase the product from 0.20 (P(B)) to 0.30 (P(B|S)). Therefore, the advertisement seems to have a positive effect on customer purchases.

As a decision maker, if the cost of the advertisement is reasonable, it would be recommended to continue the advertisement since it increases the likelihood of product purchases.

b. The company's market share can be estimated by considering the probability of individuals purchasing the company's soap product and the probability of individuals purchasing from competitors.

The probability of an individual purchasing from competitors

= 1 - P(B)

= 1 - 0.20

= 0.80

Therefore, the company's market share is:

Market Share = P(B)

                      = 0.20

Continuing the advertisement could potentially increase the company's market share since the advertisement has a positive effect on customer purchases, as shown in part (a).

c. For the other advertisement:

Given:

P(S) = 0.30

P(B∩S) = 0.10

Using the formula for conditional probability:

P(B|S) = P(B∩S) / P(S)

So, P(B|S)

= 0.10 / 0.30

= 0.333

Comparing the two advertisements, the first advertisement had a conditional purchase probability of 0.30, while the second advertisement had a conditional purchase probability of 0.333. Therefore, the second advertisement seems to have had a slightly bigger effect on customer purchases.

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The advertisements increase the chance for an individual to purchase the product. The second advertisement yields a higher likelihood of purchase, implying it's more effective.

The problem is related to conditional probability. To solve the question:

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Answer:

a. D'(p) = -10p - 6

b. There is a decrease of 96 units of demand for each dollar increase

Step-by-step explanation:

The demand function is:

[tex]D(p) = - 5p^2-6p+400[/tex]

(a) The derivate of the demand function with respect to price gives us the rate of change of demand:

[tex]\frac{dD(p)}{dp}=D'(p) = -10p-6[/tex]

(b) When p = $9, the rate of change of demand is:

[tex]D'(9) = -10*9-6\\D'(9) = -96\ \frac{units}{\$}[/tex]

This means that, when p = $9,  there is a decrease of 96 units of demand for each dollar increase.

Final answer:

The rate of change of demand with respect to price is given by the derivative D'(p) = -10p - 6. When the price is $9, the rate of change of demand is -96, indicating that for each dollar increase in price, demand decreases by 96 items.

Explanation:

To address the demand model problem, we first need to calculate the rate of change of demand with respect to price. This involves taking the derivative of the demand function D(p) = -5p^2 - 6p + 400 with respect to price p. The derivative, D'(p), is calculated as follows:

Differentiate each term with respect to p:

The derivative of -5p^2 is -10p.

The derivative of -6p is -6.

The derivative of a constant (400) is 0.

Combine these to get the rate of change formula D'(p) = -10p - 6.

To find the rate of change of demand when the price is $9, we substitute p with 9 into the rate of change formula:

D'(9) = -10(9) - 6 = -90 - 6 = -96

The rate of change of demand at a price of $9 is -96 items per dollar. This means that for each one dollar increase in price, the quantity demanded decreases by 96 items.

Answer:

the 10th term of the sequence is 3

Step-by-step explanation:

A sequence is a list of numbers or objects in a specific order

The nth term given in the question is not a function of n

i.e aₙ= 3

Since the sequence starts with an index of 1

a₁=3

All other terms in the sequence will also be 3. Meaning that it is an arithmetic progression with a common difference of 0.

The 10th term is given by

a₁₀= 3

Answer:

Step-by-step explanation:

The formula for finding the sum of the measure of the interior angles in a regular polygon is expressed as (n - 2) × 180. Therefore,

(n - 2) × 180 = 9000

180n - 360 = 9000

180n = 9000 + 360 = 9360

n = 9360/180

n = 52

The regular polygon has 52 sides

7) The sum of the angles in the quadrilateral is 360°. Let x represent the missing angle. Therefore,

64 + 116 + 120 + x = 360

300 + x = 360

x = 360 - 300

x = 60°

8a) let x represent the missing side. Therefore,

24/15 = x/10

Cross multiplying,

15x = 240

x = 240/15 = 16

8b) let x represent the missing side. Therefore,

6/12 = 5/x

6x = 60

x = 60/6 = 10

Answer: 160 liters of the

40% solution and 80 liters of the 70% solution will be used.

Step-by-step explanation:

Let x represent the number of liters of 40% antifreeze solution that should be used.

Let y represent the number of liters of 70% antifreeze solution that should be used.

The volume of the mixture to be mixed is 240 liters. It means that

x + y = 240

The 40% antifreeze solution is to be mixed with a 70% antifreeze

solution to get 240 liters of a 50% solution. This means that

0.4x + 0.7y = 0.5(240)

0.4x + 0.7y = 120 - - - - - - - - - - - -1

Substituting x = 240 - y into equation 1, it becomes

0.4(240 - y) + 0.7y = 120

96 - 0.4y + 0.7y = 120

- 0.4y + 0.7y = 120 - 96

0.3y = 24

y = 24/0.3

y = 80

x = 240 - y = 240 - 80

x = 160

[tex]160 \ litres[/tex] of [tex]40 \%[/tex] antifreeze solution and [tex]80 \ litres[/tex] of [tex]70 \%[/tex] antifreeze solutions will be used.

Given, two solutions namely [tex]40 \%[/tex] antifreeze and [tex]70 \%[/tex] antifreeze solutions.

Let [tex]x[/tex] litres of the [tex]40 \%[/tex] antifreeze solution and [tex]y[/tex] litres of the [tex]70 \%[/tex] antifreeze solutions will be used.

Total volume of the solution,

[tex]x+y=240..........(1)[/tex]

Now, [tex]40\%[/tex] of antifreeze solution is to be mixed with a [tex]70 \%[/tex] of antifreeze

solution to get 240 liters of a [tex]50 \%[/tex] solution,

[tex]0.4x+0.7y=240\times 0.5[/tex]

[tex]0.4x+0.7y=120........(2)[/tex]

From Equation (1)  [tex]y=240-x[/tex], substitute the value of [tex]y[/tex] in Equation (2),we get

[tex]0.4x+0.7(240-x)=120\\0.4x+168-0.7x=120\\-0.3x=120-168\\-0.3x=-48\\x=160[/tex]

Putting the value of [tex]x=160[/tex], we get

[tex]y=240-160[/tex]

[tex]y=80[/tex].

Hence [tex]160 \ litres[/tex] of [tex]40 \%[/tex] antifreeze solution and [tex]80 \ litres[/tex] of [tex]70 \%[/tex] antifreeze solutions will be used.

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Answer:

The answer to your question is 3.35 ft

Step-by-step explanation:

Data

height = 19 ft

angle = 80°

Process

1.- It is formed a right triangle so use a trigonometric function that relates the opposite side and the adjacent side. This trigonometric function is tangent.

               tan Ф = Opposite side/adjacent side

               adjacent side = Opposite side / tan Ф

               adjacent side = 19 / tan 80

               adjacent side = 19 / 5.67

               adjacent side = 3.35 ft

Answer: The base of the latter needs to be positioned 3.6 feet out from the building.

Step-by-step explanation:

The ladder forms a right angle triangle with the building and the ground. The length of the ladder represents the hypotenuse of the right angle triangle. The height from the where the top of the ladder touches the window to the base of the building represents the opposite side of the right angle triangle.

The distance from the bottom of the ladder to the base of the building represents the adjacent side of the right angle triangle.

To determine distance,h from the bottom of the ladder to the base of the building, we would apply

the tangent trigonometric ratio.

Tan θ = opposite side/adjacent.

Tan 80 = 19/h

h = 19/Tan 80 = 19/5.6713

h = 3.6 feet

Answer:

Part a: The two events are termed as disjoint because the event B clearly rules out the obese person

Part b: In the plain language, the event "A or B" means that the person is  overweight or obese. Its probability is 0.74.

Part c: If C is the event that the person chosen has normal weight or less, its probability is 0.26.

Step-by-step explanation:

As per the question obtained from the google search, the question has 3 parts as follows:

Part a

Explain why events A and B are disjoint.

Solution

The two events are termed as disjoint because the event B clearly rules out the obese person so the events are disjoint. so the correct option as given in the complete question is A.

Part b

Say in plain language what the event "A or B" is.

What is P(A or B)? (Enter your answer to two decimal places.)

Solution

In the plain language, the event "A or B" means that the person is  overweight or obese. The correction option as given in the complete question is a.

P(A or B) is given as

P(A or B)=P(AUB)=P(A)+P(B)-P(A∩B)

Here from the data of

P(A)=0.41

P(B)=0.33

P(A∩B)=0 (As the events are disjoint)

P(A or B)=P(AUB)=0.41+0.33-0

P(AUB)=0.74

So the probability of A or B is 0.74.

Part c

If C is the event that the person chosen has normal weight or less, what is

P(C)? (Enter your answer to two decimal places.)

Solution

P(C) is given as

P(C)=1-P(AUB)

P(C)=1-0.74

P(C)=0.26

So the probability of event C is 0.26.

Answer: 60

Step-by-step explanation:

Given : The number of choices for different temperatures = 3

The number of choices for different pressures = 4

The number of choices for different catalyst = 5

Since , the experimenter is studying the effects of temperature, pressure, and different type of catalysts.

Then by using the Fundamental principle of counting , we have

The number of  experimental runs are available = (number of choices for  temperatures ) x (number of choices for pressures) x( number of choices for  catalyst)

= 3 x 4 x 5 = 60

Hence, the number of experimental runs are available = 60

Answer:

See explanation below.

Step-by-step explanation:

When we want to fit a linear model given by:

[tex] y = \beta_0 + \beta_1 x[/tex]

Where y is a vector with the observations of the dependent variable, [tex]\beta_0 , \beta_1 [/tex] the parameters of the model and x the vector with the observations of the independent variable.

For this case this population regression function represent the conditional mean of the variable Y with values of X constant. And since is a population regression the parameters are not known, for this reason we use the sample data to obtain the sample regression in order to estimate the parameters of interest [tex] \beta_0, \beta_1[/tex]

We can use any method in order to estimate the parameters for example least squares minimizing the difference between the fitted and the real observations for the dependenet variable.  After we find the estimators for the regression model then we have a model with the estimated parameters like this one:

[tex] \hat y = \hat b_0 +\hat b_1 x[/tex]

With [tex] \hat \beta_0 = b_o , \hat \beta_1 = b_1[/tex]

And this model represent the sample regression function, and this equation shows to use the estimated relation between the dependent and the independent variable.

Answer:

362880

Step-by-step explanation:

Given that a sales representative must visit nine cities. There are direct air connections between each of the cities

Since there are direct connections between any two pairs the sales rep can visit in any order as he wishes.

He has 9 ways to select first city, now remaining cities are 8.  He can visit any one in 8 ways.

i.e. No of ways of visiting 9 cities in any order = 9P9

= 9!

= 362880

So no of ways he visits the cities since there are direct connections between any two cities is

362880

This is a permutation problem. The sales representative has 9 factorial (9*8*7*6*5*4*3*2*1 = 362,880) different choices for the order to visit the cities.

The subject of this question is a part of combinatorics, specifically Permutations. In this scenario, the sales representative has 9 different cities to visit and the order in which the cities are visited is important.

Using the multiplication rule of counting, the number of ways he can visit these cities is 9 factorial (9!). In general, the abbreviation 'n!' denotes the product of all positive integers less than or equal to n.

So for 9 cities this would be: 9*8*7*6*5*4*3*2*1 = 362,880 different choices for order to visit the cities.

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Answer:

I think 25000

Step-by-step explanation:

Answer:

the Fact that 1610 responses where gotten from the original population of

241 500 000 makes this a convenience sampling.

Step-by-step explanation:

convenience Sampling : this is a type of non-probability sampling that involves the sample being drawn from that part of the population that is close to hand.

The histogram is shown below.

You'll have 10 bars. Under each bar is a label from 0 to 9. The height of each bar represents the frequency of each units digit.

The histogram shows two bars that are relatively large compared to the rest. These bars have frequency of 16 and 14 (for units digits of 0 and 5 respectively). The distribution is nearly bimodal. If the two frequencies mentioned were the same, say both 16, then it would be exactly bimodal. As you can probably guess, bimodal means "two modes".

The rest of the bar heights are nearly the same, so the remaining portion of the distribution is nearly uniform. A uniform distribution has every bar the same height. Collectively, all the bars of a uniform distribution forms a larger rectangle.

Answer:

6 fl. oz

Step-by-step explanation:

The most accurate measurement for the volume a teacup can hold is 6 fluid ounces.

To determine the most accurate measurement for the amount that a teacup can hold, we need to consider the typical volume of a teacup and understand the relationship between different units of capacity. A standard teacup holds about 6 fluid ounces. Given the unit conversion factors, we know that:

1 cup = 8 fluid ounces

1 pint = 2 cups = 16 fluid ounces

1 quart = 2 pints = 32 fluid ounces

1.5 quarts = 48 fluid ounces

Considering these units:

6 fluid ounces is less than 1 cup (8 fl oz).

2 cups equal 16 fluid ounces, which is more than a typical teacup can hold.

1 pint (16 fluid ounces) is also more than what a teacup can hold.

1.5 quarts (48 fluid ounces) is significantly more than a teacup's capacity.

Therefore, the most accurate measurement to describe the amount a teacup can hold is 6 fluid ounces.

Answer:

813 will exceed 9 months.

Step-by-step explanation:

For each women, there are only two possible outcomes. Either they will exceed the gestation period, or they will not. So we use the binomial probability distribution to solve this problem.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

In this problem, we have that:

[tex]n = 3000, p = 0.271[/tex]

In 3000 human gestation​ periods, roughly how many will exceed 9​ months?

[tex]E(X) = np = 3000*0.271 = 813[/tex]

813 will exceed 9 months.

The gestation period should be exceed 9 month is 813.

Given that,

Based on the above information, the calculation is as follows:

[tex]= 0.271 \times 3,000[/tex]

= 813

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Answer:

Therefore the measurement of EF,

[tex]EF=1.98\ units[/tex]

Step-by-step explanation:

Given:

In Right Angle Triangle DEF,

m∠E=90°

m∠D=26°

∴sin 26 ≈ 0.44

DF = Hypotenuse = 4.5    

To Find:

EF = ? (Opposite Side to angle D)

Solution:

In Right Angle Triangle DEF, Sine Identity,

[tex]\sin D= \dfrac{\textrm{side opposite to angle D}}{Hypotenuse}\\[/tex]

Substituting the values we get

[tex]\sin 26= \dfrac{EF}{DF}=\dfrac{EF}{4.5}[/tex]

Also,  sin 26 ≈ 0.44 .....Given

[tex]EF = 4.5\times 0.44=1.98\ units[/tex]

Therefore the measurement of EF,

[tex]EF=1.98\ units[/tex]

Answer:

By looking at the histogram, we can conclude that the distribution is unimodal and skewed to the right. The modal value lies around 30 to 31 minutes and most of the running times range between 29 and 32 minutes.

The histogram is skewed to the right with most of the outliers present at the higher running times because practically, it is most likely possible for a person to run slow and take more time to run 4 miles rather than run fast and take less time.

Step-by-step explanation:

We say that the distribution is unimodal because there is only one peak which is the highest.

The distribution is skewed to the right because most of the outliers are present at the left side of the peak.

Final answer:

a. The probability of drawing a king of any suit is 1/13. b. The probability of drawing a face card that is also a spade is 3/26. c. The probability of drawing a card without a number is 4/13. d. The probability of drawing a red card is 1/2. The probability of drawing an ace is 1/13. The probability of drawing a red ace is 1/26.

Explanation:

a. There are 4 kings in a deck of cards, one for each suit. So, the probability of drawing a king of any suit is the number of king cards divided by the total number of cards in the deck: 4/52 = 1/13 = 0.077 or 7.7%.

b. There are 3 face cards in each suit, and there are 2 black suits (spades and clubs). So, the probability of drawing a face card that is also a spade is the number of face cards (3) multiplied by the number of black suits (2), divided by the total number of cards in the deck: (3 * 2)/52 = 6/52 = 3/26 = 0.115 or 11.5%.

c. A card without a number refers to a face card (jack, queen, or king) or an ace. There are 12 face cards and 4 aces in a deck. So, the probability of drawing a card without a number is the number of face cards plus the number of aces divided by the total number of cards in the deck: (12 + 4)/52 = 16/52 = 4/13 = 0.308 or 30.8%.

d. There are 26 red cards in a deck (hearts and diamonds) and 52 total cards. So, the probability of drawing a red card is the number of red cards divided by the total number of cards: 26/52 = 1/2 = 0.5 or 50%. The probability of drawing an ace is 4/52 = 1/13 = 0.077 or 7.7%. The probability of drawing a red ace is the number of red aces divided by the total number of cards: 2/52 = 1/26 = 0.038 or 3.8%.

These events are mutually exclusive because a card cannot be an ace and also be a non-ace card at the same time. However, they are not independent because the probability of drawing a red ace would change if an ace had already been drawn.

Two events that are mutually exclusive are drawing a spade and drawing a heart. You cannot draw a card that is both a spade and a heart at the same time.

Answer:

1) 41.67% probability that dice A is larger than dice B.

2) Given hat the roll resulted in a sum of 5 or less, there is a 20% conditional probability that the two dice were equal.

3) Given that the two dice land on different numbers there is a 26.67% conditional probability that the two dice differed by 2.

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this problem, we have these possible outcomes:

Format(Dice A, Dice B)

(1,1), (1,2), (1,3), (1,4), (1,5),(1,6)

(2,1), (2,2), (2,3), (2,4), (2,5),(2,6)

(3,1), (3,2), (3,3), (3,4), (3,5),(3,6)

(4,1), (4,2), (4,3), (4,4), (4,5),(4,6)

(5,1), (5,2), (5,3), (5,4), (5,5),(5,6)

(6,1), (6,2), (6,3), (6,4), (6,5),(6,6)

There are 36 possible outcomes.

1) Find the probability that dice A is larger than dice B.

Desired outcomes:

(2,1)

(3,1), (3,2)

(4,1), (4,2), (4,3)

(5,1), (5,2), (5,3), (5,4)

(6,1), (6,2), (6,3), (6,4), (6,5)

There are 15 outcomes in which dice A is larger than dice B.

There are 36 total outcomes.

So there is a 15/36 = 0.4167 = 41.67% probability that dice A is larger than dice B.

2) Given that the roll resulted in a sum of 5 or less, find the conditional probability that the two dice were equal.

Desired outcomes:

Sum of 5 or less and equal

(1,1), (2,2)

There are 2 desired outcomes

Total outcomes:

Sum of 5 or less

(1,1), (1,2), (1,3), (1,4)

(2,1), (2,2), (2,3)

(3,1), (3,2)

(4,1)

There are 10 total outcomes.

So given hat the roll resulted in a sum of 5 or less, there is a 2/10 = 20% conditional probability that the two dice were equal.

3) Given that the two dice land on different numbers, find the conditional probability that the two dice differed by 2.

Desired outcomes

Differed by 2

(1,3), (2,4), (3,1), (3,5),(4,2),(4,6), (5,3), (6,4).

There are 8 total outcomes in which the dices differ by 2.

Total outcomes:

There are 30 outcomes in which the two dice land of different numbers.

So given that the two dice land on different numbers there is a 8/30 = 0.2667 = 26.67% conditional probability that the two dice differed by 2.

To find the angle of intersection (θ) between the curves r1(t) and r2(t) at the origin, we calculate the dot product of their tangent vectors and use the arccosine formula. θ ≈ 79 degrees.

To find the angle of intersection (θ) between the curves r1(t) = (3t, [tex]t^2[/tex], [tex]t^4[/tex]) and r2(t) = (sin(t), sin(2t), 5t) at the origin, we can use the dot product formula for angles between vectors.

First, we need to calculate the tangent vectors at the origin for both curves. The tangent vector for r1(t) is (3, 2t, [tex]4t^3[/tex]), and for r2(t), it is (cos(t), 2cos(2t), 5).

Next, evaluate these vectors at t = 0 (the origin) to get the tangent vectors at the point of intersection: r1'(0) = (3, 0, 0) and r2'(0) = (1, 2, 5).

Now, calculate the dot product of these vectors:

r1'(0) · r2'(0) = (3 × 1) + (0 × 2) + (0 × 5) = 3.

The magnitude of r1'(0) is [tex]\sqrt{ (3^2 + 0^2 + 0^2)[/tex] = 3, and the magnitude of r2'(0) is [tex]\sqrt{(1^2 + 2^2 + 5^2[/tex]) = [tex]\sqrt{(1 + 4 + 25)[/tex] = √30.

Now, use the dot product formula for angles:

cos(θ) = (r1'(0) · r2'(0)) / (|r1'(0)| ×|r2'(0)|)

cos(θ) = 3 / (3 × [tex]\sqrt{30}[/tex]) = 1 / [tex]\sqrt30}[/tex]

Now, find θ:

θ = arc cos(1 / [tex]\sqrt{30[/tex])

Using a calculator, θ ≈ 79 degrees (rounded to the nearest degree).

So, the angle of intersection θ is approximately 79 degrees.

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Answer:

0.9688

Step-by-step explanation:

For each time the coin is tossed, there are only two possible outcomes. Either it is heads, or it is tails. The probabilities for each coin toss are independent from each other. So we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

In this problem we have that:

For each time the coin is tossed, heads or tails are equally as likely, since the coin is fair. So [tex]p = \frac{1}{2} = 0.5[/tex]

You toss a fair coin 5 times. What is the probability of at least one head?

Either there are no heads, or there is at least one head. The sum of the probabilities of these events is decimal 1. So

[tex]P(X = 0) + P(X \geq 1) = 1[/tex]

We want to find [tex]P(X \geq 1)[/tex], when [tex]n = 5[/tex].

So

[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{5,0}.(0.5)^{0}.(0.5)^{5} = 0.03125[/tex]

[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.03125 = 0.96875[/tex]

Rounding to the nearest ten-thousandth(four decimal places), this probability is 0.9688.

The probability of getting at least one head when tossing a fair coin 5 times is 31/32 or 0.96875 when rounded to the nearest ten-thousandth.

The subject of this question is probability, a field within Mathematics. To answer your question: the probability of getting at least one head when tossing a fair coin 5 times can be found by calculating the probability of not getting a head (which is tossing tails 5 times in a row) and then subtracting that from 1 (representing certainty). Each toss of the fair coin has two outcomes, heads or tails, with equal probability of 1/2. If you toss the coin 5 times, the total number outs comes is 2^5, or 32. The chance of getting all tails is (1/2)^5, which is 1/32.

So, the probability of not getting a head (only tails) in 5 tosses is 1/32. Subtract this from 1 to find the probability of getting at least one head: This equals 1 - 1/32 = 31/32 = 0.96875, when rounded to the nearest ten-thousandth.

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Answer: It will take 11.56 hours .

Step-by-step explanation:

Exponential growth in population or size formula :

[tex]P(t)=P_0e^{rt}[/tex]

, where [tex]P_0[/tex] = initial size

r= rate of growth

t= time period

As per given , we have

[tex]P_0=10[/tex] grams

At t= 1 , P(t)= 11 grams

Then,

[tex]11=10e^{r(1)}\\\\ 1.1= e^r\\\\\text{Taking natural log on both sides , we get} \\\\\ln (1.1)=r\ln (e)\\\\ r=\ln (1.1)\\\\ r=0.0953101798043\approx0.095[/tex]

When, the  bacteria have tripled in size , P(t) = 3 x10 = 30

Then,

[tex]30=10e^{0.095t}\\\\ 3=e^{0.095t}[/tex]

[tex]\text{Taking natural log on both sides , we get}\\\\ \ln 3=0.095t\\\\ t=\dfrac{\ln3}{0.095}\\\\ t=\dfrac{1.09861228867}{0.095}\approx11.56[/tex]

Hence, it will take 11.56 hours .

A bacteria culture is initially 10 grams at t=0 hours & grows at a rate proportional to its size , After an hour the bacteria culture weighs 11 grams , The bacteria takes 11.56 hours to have tripled in size.

To find the time of bacteria when increasing the growth to tripled.

Given :    when time=0 hours , weight=10 grams.

               when time=1  hours , weight=11 grams.

To find:   when time= ? hours , weight=30grams.

Here according to question, initial size = 10 grams we have asked for tripled in size i.e. 30 grams.

Now we knows that,

The formula for exponential growth in population or size is

              [tex]\rm (P)=P_0e^{rt}[/tex]  where,

               [tex]\rm P_0=initial\;size\\\\r= rate\;of\;growth\\\\t= time \;period[/tex]

Now, we put the value in formula we get,

[tex]\rm P_0=10\;grams \\\\when ,\\\;\;t=1\;hour P(t)=11 grams\\Then,\\11=10e^{r(1)\\1.1 =e^r\\\\\rm Taking \;log(natural)\;both\;the\; side \;on \;solving\;we\;get,\\ln(1.1)=r\;ln(e)\\r=ln(1.1)\\r=0.953101798043\approx0.095[/tex]

Now when the bacteria increase its size to triple

[tex]\rm P(t) = 3 \times 10 = 30[/tex]

Then, according to the formula we substitute values in the formula,

[tex]\rm 30=10e^{0.095t}\\\\3=e^{0.095t}\\\\Again \;we \;take\;natural\;log\;on \;both\;the\;sides, we\;get\\ln\;3=0.095t\\\\t=\dfrac{\rm ln\;3}{0.095}\\\\\\\\\rm t= \dfrac{1.09861228867}{0.095} \\\\\ t=approx \; 11.56[/tex]

Therefore, The bacteria takes 11.56 hours to have tripled in size.

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Answer:

a) 6feet/secs

b) 6feet/secs

c) 6feet/secs

Step-by-step explanation:

The detailed steps are as shown in the attachment

The average speed of the car in each time interval is calculated by first evaluating the distance formula at the endpoints of the interval, subtracting to find the distance travelled, and then dividing by the time taken to travel that distance.

The given formula is

d = 6t - 11

, where 'd' is the distance in feet, and 't' is the time in seconds since the car started moving. Firstly, to find the average speed, which is the distance travelled divided by time taken, we need to calculate the distance travelled in each interval. For instance, for the interval from 't=3' to 't=6', we first calculate the distances 'd' at t=3 and t=6 by substituting them into the equation, then subtracting the two to get the distance travelled over this time interval. Similarly, the distances travelled in the intervals from t=6 to t=6.5 seconds and t=6.5 to t=7 seconds were calculated. Finally, the

average speed

in each time interval is obtained by dividing that interval's travelled distance by the time taken.

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Answer:

A number to the third power would be a number cubed, so the answer is "Cubed".

Answer:

please see answers are as in the explanation.

Step-by-step explanation:

As from the data of complete question,

[tex]0\leq x\leq 1\\0\leq y\leq 1\\0\leq z\leq 1\\u= \alpha x\\v=\beta y\\w=0[/tex]

The question also has 3 parts given as

Part a: Sketch the deformed shape for α=0.03, β=-0.01 .

Solution

As w is 0 so the deflection is only in the x and y plane and thus can be sketched in xy plane.

the new points are calculated as follows

Point A(x=0,y=0)

Point A'(x+αx,y+βy)

Point A'(0+(0.03)(0),0+(-0.01)(0))

Point A'(0,0)

Point B(x=1,y=0)

Point B'(x+αx,y+βy)

Point B'(1+(0.03)(1),0+(-0.01)(0))

Point B'(1.03,0)

Point C(x=1,y=1)

Point C'(x+αx,y+βy)

Point C'(1+(0.03)(1),1+(-0.01)(1))

Point C'(1.03,0.99)

Point D(x=0,y=1)

Point D'(x+αx,y+βy)

Point D'(0+(0.03)(0),1+(-0.01)(1))

Point D'(0,0.99)

So the new points are A'(0,0), B'(1.03,0), C'(1.03,0.99) and D'(0,0.99)

The plot is attached with the solution.

Part b: Calculate the six strain components.

Solution

Normal Strain Components

                             [tex]\epsilon_{xx}=\frac{\partial u}{\partial x}=\frac{\partial (\alpha x)}{\partial x}=\alpha =0.03\\\epsilon_{yy}=\frac{\partial v}{\partial y}=\frac{\partial ( \beta y)}{\partial y}=\beta =-0.01\\\epsilon_{zz}=\frac{\partial w}{\partial z}=\frac{\partial (0)}{\partial z}=0\\[/tex]

Shear Strain Components

                             [tex]\gamma_{xy}=\gamma_{yx}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=0\\\gamma_{xz}=\gamma_{zx}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}=0\\\gamma_{yz}=\gamma_{zy}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}=0[/tex]

Part c: Find the volume change

[tex]\Delta V=(1.03 \times 0.99 \times 1)-(1 \times 1 \times 1)\\\Delta V=(1.0197)-(1)\\\Delta V=0.0197\\[/tex]

Also the change in volume is 0.0197

For the unit cube, the change in terms of strains is given as

             [tex]\Delta V={V_0}[(1+\epsilon_{xx})]\times[(1+\epsilon_{yy})]\times [(1+\epsilon_{zz})]-[1 \times 1 \times 1]\\\Delta V={V_0}[1+\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}+\epsilon_{xx}\epsilon_{yy}+\epsilon_{xx}\epsilon_{zz}+\epsilon_{yy}\epsilon_{zz}+\epsilon_{xx}\epsilon_{yy}\epsilon_{zz}-1]\\\Delta V={V_0}[\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\[/tex]

As the strain values are small second and higher order values are ignored so

                                      [tex]\Delta V\approx {V_0}[\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\ \Delta V\approx [\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\[/tex]

As the initial volume of cube is unitary so this result can be proved.

Answer:

a) F(t) = R[P(t)]

b) the output units of the new function = F(t) in dollars per gallon

Step-by-step explanation:

a) There are two function R(t) which shows the average price in dollars of a gallon of regular unleaded gasoline and P(t) which shows the purchasing power of the dollar as measured by consumer prices based on 2010 dollars.

To write the function which gives the rice of gasoline in constant 2010 dollars ;

From the analysis , this is an example of a composition of function as such the relationship =

F(t) = R[P(t)]

b) the output units of the new function = F(t) in dollars per gallon

This shows that the value of F(t) is the dependent variable

Final answer:

To determine the real price of gasoline in constant 2010 dollars, combine the functions R(t) and P(t) using the formula R(t) ÷ P(t). This calculation adjusts the nominal price of gasoline for inflation, resulting in the price of gasoline in terms of 2010 dollars, with the output units being 2010 dollars per gallon.

Explanation:

To combine the two functions representing the average price of a gallon of regular unleaded gasoline, R(t), and the purchasing power of the dollar as measured by consumer prices based on 2010 dollars, P(t), into a new function giving the price of gasoline in constant 2010 dollars, we use the formula:

R(t) ÷ P(t)

This formula represents the price of gasoline adjusted for inflation, giving us the real price of gasoline in terms of 2010 dollars. Here, R(t) gives the average price of gasoline in year t, and P(t) gives the purchasing power of the dollar in year t, compared to 2010 dollars. By dividing R(t) by P(t), we adjust the nominal price of gasoline to reflect its real value, accounting for changes in the purchasing power of the dollar over time.

The output units of this new function would be 2010 dollars per gallon. This metric allows economists and analysts to compare the price of gasoline across different years on a level playing field, eliminating the effects of inflation.

Answer with Step-by-step explanation:

We are given that

[tex]yx+9y=0[/tex]

[tex]-4x+5y=3[/tex]

We have to find the angle between the lines.

[tex]y(x+9)=0[/tex]

[tex]y=0,x+9=0\implies x=-9[/tex]

[tex]y=0[/tex]..(1)

[tex]x=-9[/tex]..(2)

[tex]-4x+5y=3[/tex]..(3)

The angle between two lines

[tex]a_1x+b_1y+c_1=0[/tex]

[tex]a_2x+b_2y+c_2=0[/tex]

[tex]tan\theta=\mid \frac{a_1b_2-b_1a_2}{a_1a_2+b_1b_2}\mid[/tex]

By using the formula the angle between equation (1) and equation (2) is given by

[tex]tan\theta_1=\mid\frac{0\times 0-1\times 1}{0+0}\mid=\infty=90^{\circ}[/tex]degree

[tex]tan90^{\circ}=\infty[/tex]

It is not possible because we are given that the acute angle between intersecting lines that do not cross at right angles is same as the angle determined by vectors that either normal to the lines or parallel to lines.

By using the formula the angle between equation (2) and equation(3)

[tex]tan\theta_2=\mid\frac{1(5)-0(4)}{-4(1)+5(0)}\mid=\frac{5}{4}[/tex]

[tex]\theta_2=tan^{-1}(1.25)[/tex] degree

By using the formula the angle between equation (3) and equation(1)

[tex]tan\theta_3=\mid\frac{-4(1)-5(0)}{-4(0)+5(1)}\mid=\frac{4}{5}[/tex]

[tex]\theta_3=tan^{-1}(\frac{4}{5})[/tex]degree

"determine the location" or namely, is it inside the circle, outside the circle, or right ON the circle?

well, we know the center is at (1,-5) and it has a radius of 5, so the distance from the center to any point on the circle will just be 5, now if (4,-1) is less than that away, is inside, if more than that is outiside and if it's exactly 5 is right ON the circle.

well, we can check by simply getting the distance from the center to the point (4,-1).

[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ \stackrel{center}{(\stackrel{x_1}{1}~,~\stackrel{y_1}{-5})}\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{-1})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d = \sqrt{[4-1]^2+[-1-(-5)]^2}\implies d=\sqrt{(4-1)^2+(-1+5)^2} \\\\\\ d = \sqrt{3^2+4^2}\implies d =\sqrt{9+16}\implies d=\sqrt{25}\implies \stackrel{\textit{right on the circle}}{d = 5}[/tex]

Answer:

a) [tex] \hat p =\frac{X}{n}=\frac{1760}{2000}=0.88[/tex]

b) [tex]ME=1.64* \sqrt{\frac{0.88*(1-0.88)}{2000}}=0.0119[/tex]

c) [tex]0.88 - 1.64 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.868[/tex]

[tex]0.88 + 1.64 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.892[/tex]

And the 90% confidence interval would be given (0.868;0.892).

d) [tex]0.88 - 1.96 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.8658[/tex]

[tex]0.88 + 1.96 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.8942[/tex]

And the 90% confidence interval would be given (0.8658;0.8942).

Step-by-step explanation:

Part a

For this case the point of estimate for the population proportion is given by:

[tex] \hat p =\frac{X}{n}=\frac{1760}{2000}=0.88[/tex]

Part b

The confidence interval would be given by this formula

[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

For the 90% confidence interval the value of [tex]\alpha=1-0.9=0.1[/tex] and [tex]\alpha/2=0.05[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.

[tex]z_{\alpha/2}=1.64[/tex]

The margin of error is given by:

[tex] ME=z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

And if we replace we got:

[tex]ME=1.64* \sqrt{\frac{0.88*(1-0.88)}{2000}}=0.0119[/tex]

Part c

And replacing into the confidence interval formula we got:

[tex]0.88 - 1.64 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.868[/tex]

[tex]0.88 + 1.64 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.892[/tex]

And the 90% confidence interval would be given (0.868;0.892).

Part d

For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.

[tex]z_{\alpha/2}=1.96[/tex]

[tex]0.88 - 1.96 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.8658[/tex]

[tex]0.88 + 1.96 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.8942[/tex]

And the 90% confidence interval would be given (0.8658;0.8942).

The Main Answer for:

a. The point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern is approximately [tex]0.88[/tex].

b. The margin of error at [tex]90[/tex]% confidence is approximately [tex]0.0120[/tex].

c. The [tex]90[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern is approximately ([tex]0.868, 0.892[/tex]).

d. The [tex]95[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern is approximately ([tex]0.8657, 0.8943[/tex]).

a. Point Estimate: The point estimate of the population proportion can be calculated by dividing the number of respondents who think the future health of Social Security is a major economic concern by the total number of respondents surveyed.

Point Estimate = Number of respondents concerned about Social Security / Total number of respondents

Given: Number of respondents concerned about Social Security = [tex]1760[/tex]

Total number of respondents surveyed = [tex]2000[/tex]

Point Estimate = [tex]1760 / 2000 =0.88[/tex]

So, the point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern is approximately [tex]0.88[/tex].

b. Margin of Error: The margin of error (E) can be calculated using the formula:

[tex]E=z*\sqrt{p(1-p)/n}[/tex]

where:

• z is the z-score corresponding to the desired confidence level

• p is the point estimate of the population proportion

• n is the sample size

Since we are aiming for a 90% confidence interval, we find the z-score corresponding to a 90% confidence level, which is approximately 1.645 (you can find this value in a standard normal distribution table).

[tex]E=1.645*\sqrt{0.88(1-0.88)/ 2000} \\E=1.645*\sqrt{0.88*0.12/2000} \\E=1.645*\sqrt{0.1056/2000} \\E=1.645*\sqrt{0.0000528} \\E=1.645*0.00727\\E=0.01196[/tex]

So, the margin of error at [tex]90[/tex]% confidence is approximately [tex]0.0120[/tex].

c. Confidence Interval (90%): The confidence interval can be calculated using the point estimate and the margin of error.

[tex]CI=(p-E,p+E)\\CI=(0.88-0.0120,0.88+0.0120)\\CI=(0.868,0.892)[/tex]

So, the [tex]90[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern is approximately ([tex]0.868, 0.892[/tex]).

d. Confidence Interval (95%): To calculate the 95% confidence interval, we use the same formula but with a different z-score. For a 95% confidence level, the z-score is approximately 1.96.

[tex]E=1.96*\sqrt{0.88(1-0.88)/ 2000} \\E=1.96*\sqrt{0.0000528} \\E=1.96*0.00727\\E=0.01427\\CI=(0.88-0.0143,0.88+0.0143)\\CI=(0.8657,0.8943)\\[/tex]

So, the [tex]95[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern is approximately ([tex]0.8657, 0.8943[/tex]).

COMPLETE QUESTION:

The Consumer Reports National Research Center conducted a telephone survey of [tex]2000[/tex] adults to learn about the major economic concerns for the future (Consumer Reports, January [tex]2009[/tex]). The survey results showed that [tex]1760[/tex] of the respondents think the future health of Social Security is a major economic concern.

a. What is the point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern (to 2 decimals)?

b. At [tex]90[/tex]% confidence, what is the margin of error (to 4 decimals)?

c. Develop a [tex]90[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern (to 3 decimals).(, )

d. Develop a [tex]95[/tex]% confidence interval for this population proportion (to 4 decimals).

Final answer:

The probability that at least two bills must be selected to obtain the first $10 bill is approximately 24.7%.

Explanation:

To find the probability that at least two bills must be selected to obtain the first $10 bill, we need to calculate the probability of not drawing a $10 bill on the first draw and then drawing a $10 bill on the second draw. In total, there are 5 + 3 + 6 = 14 bills in the wallet.

On the first draw, the probability of not getting a $10 bill is the number of non-$10 bills over the total number of bills, which is (3 $5 bills + 6 $1 bills) / 14 total bills = 9/14.

Assuming a non-$10 bill was drawn first, there are now 13 bills left in the wallet. The probability of drawing a $10 bill on the second draw is now the number of $10 bills remaining over the total number of bills left, which is 5/13.

The combined probability of these two events happening in sequence (not drawing a $10 bill first and then drawing a $10 bill) is the product of their probabilities: (9/14) * (5/13).

Thus, the total probability is (9/14) * (5/13) = 45/182, which simplifies to approximately 0.247 or 24.7%.