The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modification of (2), given in Section 3.1, is dT dt = kS(T − Tm), where k < 0 and Tm is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 145° F. The exposed surface area of the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the coffee in cup A is 95° F. If Tm = 65° F, then what is the temperature of the coffee in cup B after 30 min? (Round your answer to two decimal places.)

Answer:

3/20

Step-by-step explanation:

[tex]\frac{3}{5}[/tex]  ÷  4    =     [tex]\frac{3}{5}[/tex] [tex]* \frac{1}{4}[/tex]    =      [tex]\frac{3 * 1}{5 * 4}[/tex]     =     [tex]\frac{3}{20}[/tex]

To calculate the fraction of the class that earned an A, multiply 3/5 by 1/4, which equals 3/20. Thus, 3/20 of the class earned an A. This is the correct answer among the given options.

Calculating the Fraction of a Class That Earned an A

To find out what fraction of the class earned an A, we start with the fact that 3/5 of the class earned an A or a B.

We also know that 1/4 of those students earned an A.

To determine the fraction of the entire class that earned an A, we multiply these two fractions together:

Fraction of class that earned an A = (3/5) × (1/4)

This multiplication gives us:

(3 × 1) / (5 × 4) = 3/20

Therefore, 3/20 of the class earned an A.

This fraction cannot be reduced further and it is also one of the options provided, confirming that it is the correct answer.

Answer:

a) 0.659

b) 0.341

c) It'll be smarter to prepare to defend the left hand side as the probability of play going through that side when the right guard takes a balanced stance is almost double of the probability that the play would go through the right when the right guard takes a balanced stance.

Step-by-step explanation:

Considering the left first

Play goes through the left 30% of the time

- The right guard takes a balanced stance 90% of the time when play goes through the left, that is, 0.9 × 0.3 = 27% overall.

- He takes a shift stance 10% of the time that play goes through the left = 0.1 × 0.3 = 3% overall.

Play goes through the right, 70% of the time.

- He takes a balanced stance 20% of the time that play goes through the right, that is, 0.7 × 0.2 = 14% overall

- He takes a shift stance 80% of the time that play goes through the right, that is, 0.7 × 0.8 = 56% overall.

Then the right guard takes a balanced stance 27% + 14% of all the time = 41% overall.

a) Probability that play goes through the left with the right guard in a balanced stance = 27/41 = 0.659

b) Probability that play goes through the right with the right guard in a balanced stance = 14/41 = 0.341

c) It'll be smarter to prepare to defend the left hand side as the probability of play going through that side when the right guard takes a balanced stance is almost double of the probability that the play would go through the right when the right guard takes a balanced stance.

Using conditional probability, we can find the probabilities of the play going left or right. The probability of the play going left is 0.3, and the probability of the play going right is 0.7. If the linebacker sees a balanced stance, the probability of the play going left is approximately 0.658, so the linebacker should prepare to defend against a play going left.

To find the probabilities of the play going left or right, we need to use conditional probability. Let's denote L as the event of the play going left, and R as the event of the play going right. We are given P(L) = 0.3 and P(R) = 0.7. We also have the following conditional probabilities: P(B|L) = 0.9 (balanced stance when play goes left) P(S|L) = 0.1 (shift stance when play goes left) P(B|R) = 0.2 (balanced stance when play goes right) P(S|R) = 0.8 (shift stance when play goes right)

a. To find the probability that the play will go left, we can use the Law of Total Probability. P(L) = P(L∩B) + P(L∩S) = P(L)P(B|L) + P(L)P(S|L) = 0.3 X 0.9 + 0.3 * 0.1 = 0.27 + 0.03 = 0.3.

b. Similarly, to find the probability that the play will go right, P(R) = P(R∩B) + P(R∩S) = P(R)P(B|R) + P(R)P(S|R) = 0.7 X 0.2 + 0.7 X 0.8 = 0.14 + 0.56 = 0.7.

c. If the linebacker sees a balanced stance, it means that P(B) = P(B|L)P(L) + P(B|R)P(R) = 0.9 X 0.3 + 0.2 X 0.7 = 0.27 + 0.14 = 0.41. Now, we can compare the conditional probability of the play going left given a balanced stance, P(L|B) = P(B|L)P(L)/P(B) = (0.9 X 0.3)/0.41 = 0.27/0.41 ≈ 0.658. Therefore, the linebacker should prepare to defend against a play going to the left.

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Answer:

[tex]z=\frac{0.18 -0.1}{\sqrt{\frac{0.1(1-0.1)}{100}}}=2.67[/tex]  

[tex]p_v =P(z>2.67)=0.0038[/tex]  

So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of adults who rate Pepsi as being "concerned with my health" i significantly higher than 0.1

Step-by-step explanation:

Data given and notation

n=100 represent the random sample taken

X=18 represent the adults who rate Pepsi as being "concerned with my health"

[tex]\hat p=\frac{18}{100}=0.18[/tex] estimated proportion of adults who rate Pepsi as being "concerned with my health"

[tex]p_o=0.10[/tex] is the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the true proportion is higher than 0.1.:  

Null hypothesis:[tex]p \leq 0.1[/tex]  

Alternative hypothesis:[tex]p > 0.1[/tex]  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.18 -0.1}{\sqrt{\frac{0.1(1-0.1)}{100}}}=2.67[/tex]  

Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The next step would be calculate the p value for this test.  

Since is a right tailed test the p value would be:  

[tex]p_v =P(z>2.67)=0.0038[/tex]  

So the p value obtained was a very low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of adults who rate Pepsi as being "concerned with my health" i significantly higher than 0.1

The student is asked to perform a hypothesis test at the  alpha = 0.05 level to determine if the percentage of participants rating Pepsi as 'concerned with my health' has statistically increased from 10% to 18% in a follow-up survey.

The question involves conducting a hypothesis test to determine if the percentage of participants rating Pepsi as being "concerned with my health" has increased after a new initiative. With an initial percentage of 10%, a follow-up survey showed that 18 out of 100 (or 18%) now rate Pepsi in this manner. To determine if this is a statistically significant increase, at the alpha = 0.05 significance level, a hypothesis test can be performed using the binomial or normal approximation to the binomial distribution.

Null hypothesis (H0): p = 0.10 (The percentage of those who believe Pepsi is concerned with their health has not changed)
Alternative hypothesis (Ha): p > 0.10 (The percentage has increased)

If the calculated p-value is less than the significance level (0.05), we can reject the null hypothesis in favor of the alternative hypothesis, thus concluding that the percentage has indeed increased.

Given:

m∠T = 61°

m∠S = 5x - 15

m ext∠TUV = 8x - 8

To find:

The measure of exterior angle TUV

Solution:

STU is a triangle.

By exterior angle of triangle property:

The measure of an exterior angle is equal to the sum of the opposite interior angles.

m ext∠TUV = m∠T + m∠S

8x - 8 = 61 + 5x - 15

8x - 8 = 46 + 5x

Add 8 on both sides.

8x - 8 + 8 = 46 + 5x + 8

8x = 54 + 5x

Subtract 5x from both sides.

8x - 5x = 54 + 5x - 5x

3x = 54

Divide by 3 on both sides.

[tex]$\frac{3x}{3} =\frac{54}{3}[/tex]

x = 18

Substitute x = 18 in m ext∠TUV.

m ext∠TUV = 8(18) - 8

                   = 144 - 8

                   = 136

Therefore measure of exterior angle ∠TUV is 136°.

Answer:

11.51% probability that the sample average will be less than 182

Step-by-step explanation:

To solve this question, we have to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]

In this problem, we have that:

[tex]\mu = 187, \sigma = 32, n = 64, s = \frac{32}{\sqrt{64}} = 4[/tex]

What is the probability that the sample average will be less than 182

This is the pvalue of Z when X = 182. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{182 - 187}{4}[/tex]

[tex]Z = -1.2[/tex]

[tex]Z = -1.2[/tex] has a pvalue of 0.1151

11.51% probability that the sample average will be less than 182

Answer:

a) y = -¼

Step-by-step explanation:

(-2x^6+5x+8)/(8x^6+6x+5)

Just divide the leading coefficients

-2/8 = -¼

Horizontal asymptote:

y = -¼

Answer:a) y = -¼

Step-by-step explanation:

(-2x^6+5x+8)/(8x^6+6x+5)

Just divide the leading coefficients

Answer:

A hypothesis is a proposed statement explaining the relationship between two or more variable which can be verifiable through experiment. In this case a testable hypothesis would be.

When yeast and sugar are mixed together during formation, it will cause the sugar to undergo a high rate of glycolysis and CO[tex]^{2}[/tex] will be present in the fermentation tube.

The arc length of AB = 8.37 meters

Solution:

Degree of AB (θ) = 60°

Radius of the circle = 8 m

Let us find the arc length of AB.

Arc length formula:

[tex]$\text{Arc length}=2 \pi r\left(\frac{\theta}{360^\circ}\right)[/tex]

                 [tex]$=2 \times 3.14 \times 8 \left(\frac{60^\circ}{360^\circ}\right)[/tex]

                 [tex]$=2 \times 3.14 \times 8 \left(\frac{1}{6}\right)[/tex]

Arc length = 8.37 m

Hence the arc length of AB is 8.37 meters.

Answer:

c. Inductive and Strong

Step-by-step explanation:

In inductive reasoning, provided data is analyzed in order to reach a conclusion. In this case, the argument provides data regarding Jane and Nancy's awards and their love for mathematics and then draws a conclusion regarding Nancy's performance in a particular class, this is an example of inductive reasoning.

As for the strength of the argument, it is plausible to infer that Jane and Nancy have similar mathematics skills since they both love calculus and excel academically. Therefore, if Jane does well in the calculus class, it is a strong argument to say that Nancy does as well.

The answer is :

c. Inductive and Strong

Final answer:

The argument is Deductive and Invalid as it does not guarantee Nancy's success in the calculus class based on Jane's performance.

Explanation:

The argument presented is Deductive and Invalid. This is because the conclusion that Nancy will do well in the calculus class based on Jane's performance does not necessarily follow logically. Just because Jane excelled does not guarantee the same result for Nancy.

In deductive reasoning, the conclusion must follow necessarily from the premises, which is not the case here. An example of a valid deductive argument would be 'All humans are mortal, Socrates is a human, therefore Socrates is mortal.'

Because the argument in question does not provide a guarantee of Nancy's success based on Jane's performance, it is categorized as Invalid in the realm of deductive reasoning.

Answer: there are 70 dimes and 40 quarters.

Step-by-step explanation:

A dime is 10 cents

A nickel is 5 cents

A quarter is 25 cents

Let x represent the number of dimes at the bottom.

Let y represent the number of quarters at the bottom.

The city park employee collected 1,850 cents in nickels, dimes, and quarters at the bottom of a wishing well. If there were 30 nickels, it means that

10x + 25y + 30 × 5 = 1850

10x + 25y = 1850 - 150

10x + 25y = 1700- - - - - - - - -1

There is a combined total of 110 dimes and quarters. It means that

x + y = 110

Substituting x = 110 - y into equation 1, it becomes

10(110 - y) + 25y = 1700

1100 - 10y + 25y = 1700

1100 + 15y = 1700

15y = 1700 - 1100

15y = 600

y = 600/15

y = 40

x = 110 - y = 110 - 40

x = 70

Final answer:

The standard deviation of the sample of numbers is approximately 10.68, after calculating the mean, finding the deviations, squaring them, averaging the squared deviations to get the variance, and then taking the square root of the variance.

Explanation:

The student is asking how to calculate the standard deviation of a sample of numbers. The sample given consists of the counts of bacteria found in ocean water, which are 41, 62, 45, 48, 69. To calculate the standard deviation, follow these steps:

Performing the calculations, you get:

Answer:

The answer to your question is Circle's area = 113.04 in²  Slice' area = 14.13 in²

Step-by-step explanation:

Data

diameter = 12 in

8 slices

Area = ?

Area of a slice = ?

Process

1.- Calculate the area of the pizza

radius = 12/2

radius = 6 in

Area = πr²

-Substitution

Area = (3.14)(6)²

-Simplification

Area = 113.04 in²

2.- Calculate the area of the slice

We can calculate this area by two methods

a) Divide the area of the number of slices

       Area of the slice = 113.04 / 8

                                   = 14.13 in²

b) Using the area of a sector

-Find the angle of each slice

               360 / 8 = 45°

-Convert 45° to rad

                180° ------------------- πrad

                  45° ------------------- x

                  x = 45πrad/180

                  x = 1/4πrad

- Calculate the area of the sector

Area = 1/4(3.14)(6)²/2

Area = 113.04/8

Area = 14.13 in²                        

Answer:

Step-by-step explanation:

Assuming the pizza is circular, we would apply the formula for determining the area of a circle. It is expressed as

Area = πr²

Where

r represents the radius of the circle.

π is a constant whose value is 3.14

From the information given,

Diameter = 12 inches

Radius = 12/2 = 6 inches

Area = 3.14 × 6² = 113.04 inches²

The formula for determining the area of a sector is expressed as

Area of sector = θ/360 × πr²

Where θ represents the central angle.

The complete circle or pizza is 360°. Since it is divided into 8 equal parts, then the central angle formed by each sector is

360/8 = 45°

Therefore,

Area of sector = 45/360 × 3.14 × 6²

= 14.13 inches²

The probability that the market research predicts an unsuccessful market for the product and the product is actually unsuccessful is 24 percent. This is calculated using the concept of conditional probability and multiplying the probability of the product actually failing (30%) with the probability of market research correctly predicting a failure (80%).

To calculate the probability that the market research indicates an unsuccessful market for the product and the product is actually unsuccessful, we first need to understand the concept of conditional probability, given certain conditions.

Here, there is a 30 percent chance that a product will fail. The market research incorrectly predicts success for failed products 20 percent of the time. Therefore, the market research predicts failure for failed products 80 percent of the time (100% - 20%).

By multiplying these probabilities together, we can find the joint probability that a product is unsuccessful and market research also predicts it to be unsuccessful.

The result is 0.30 x 0.80 = 0.24. Therefore, there is a 24 percent chance that the market research will predict a failure and the product will also actually be a failure.

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The probability that market research indicates an unsuccessful market and the product is unsuccessful is 24%.

To calculate the probability that the results indicate an unsuccessful market for the product and that the product is unsuccessful, we need to use Bayes' Theorem. Initially, we have the probability of a product being successful, P(Success) = 0.70, and the probability of a product being a failure, P(Failure) = 0.30. Furthermore, if the product is successful, market research predicts it successful 90% of the time. If the product is a failure, market research incorrectly predicts it as a success 20% of the time, meaning it correctly predicts failure 80% of the time (100% - 20%).

We are interested in P(Market Research Failure and Actual Failure). This is the probability that the market research indicates a failure and the product indeed fails. Using the probabilities provided:

Therefore, P(Market Research Failure and Actual Failure) can be calculated as:

P(Market Research Failure and Actual Failure) = P(Market Research Failure | Failure) x P(Failure) = 0.80 x 0.30 = 0.24 or 24%.

Let x* denote Peter's score. Then

P(X > x*) = 0.025

P((X - 896)/174 > (x* - 896)/174) = 0.025

P(Z > z*) = 1 - P(Z < z*) = 0.025

P(Z < z*) = 0.975

where Z follows the standard normal distribution (mean 0 and std dev 1).

Using the inverse CDF, we find

P(Z < z*) = 0.975  ==>  z* = 1.96

Then solve for x*:

(x* - 896)/174 = 1.96  ==>  x* = 1237.04

so Peter's score is roughly 1237.

Answer:

[tex]a. \ P(X\leq 8)=0.932\\b. \ P(X=8)=0.065\\c. \ P(9\leq X)=0.068\\d. \ P(5\leq X \leq 8)=0.491\\[/tex]

e. P(5<X<8)=0.251

Step-by-step explanation:

Given that boiler follows a Poisson distribution,$\sim$

[tex]X $\sim$Poi(\mu=5)[/tex]

Poisson Distribution formula is given by the expression:-

[tex]p(x,\mu)=\frac{e^{-\mu}\mu^x}{x!}[/tex]

Our probabilities will be calculated as below:

a.

[tex]P(X=x)=\frac{5^xe^{-5}}{x!}\\P(X\leq 8)=P(X=0)+P(X=1)+...+P(X=8)\\=\frac{5^0e^{-5}}{1!}+...+\frac{5^8e^{-5}}{8!}\\=0.006738+...+0.065278\\=0.961[/tex]

b.

[tex]P(X=8)=\frac{5^8e^{-5}}{8!}\\=0.065[/tex]

c.

[tex]P(9\leq X)=1-P(X=0)-P(X=1)-...-P(X=8)\\=0.068[/tex]

d.

[tex]P(5\leq X\leq 8)=P(X=5)+P(X=6)+P(X=7)+P(X=8)\\=0.491[/tex]

e.P(5<X<8)

[tex]P(X=6)+P(X=7)\\=0.251[/tex]

Answer:

Avis Company is a car rental company that is located three miles from the Los Angeles airport (LAX). Avis is dispatching a bus from its offices to the airport every 2 minutes. The average traveling time (round-trip) is 20 minutes.

a. How many Avis buses are traveling to and from the airport?

b. The branch manager wants to improve the service and suggests dispatching buses every 0.5 minutes. She argues that this will reduce the average traveling time from the airport to Avis offices to 2.5 minutes. Is she correct? If your answer is negative, then what will the average traveling time be?

The answers to the question are

a. 10 buses

b. Yes, increasing the fleet by a factor of 4 will reduce the apparent average travel time from the airport to Avid offices from 10 minutes to 2.5 minutes

Step-by-step explanation:

Number of buses dispatched every 20 minutes= 20÷2=10

Therefore 10 Avid buses are travelling to and from the Los Angeles airport

b. By dispatching every 0.5 minutes we have 20÷0.5 = 40 buses.

Since with 10 buses it takes 10 minutes to make the journey from the airport to Avid offices, then it would take 40 buses 10÷4 = 2.5 minutes time to deliver what it took 10 buses, 10 minutes to deliver.

The Avis Company has 10 buses traveling to and from LAX airport with an average round trip time of 20 minutes and a dispatch frequency of 2 minutes. Increasing the dispatch frequency to every 0.5 minutes does not reduce the average travel time; it remains at 20 minutes as it is dependent on the route's distance and traffic conditions, not the dispatch frequency.

Part A: Number of Avis Buses

To calculate the number of buses traveling to and from the airport, we use the average traveling time for a round trip and the dispatch frequency. With a round trip time of 20 minutes and buses leaving every 2 minutes, we divide the round trip time by the dispatch interval:

Total Buses = Round Trip Time / Dispatch Frequency

Total Buses = 20 minutes / 2 minutes

Total Buses = 10

Therefore, Avis has 10 buses traveling to and from the airport.

Part B: Average Traveling Time with Increased Dispatch Frequency

Increasing the dispatch frequency to every 0.5 minutes does not affect the average traveling time for a round trip. The travel time is determined by the distance between the Avis office and the airport and the traffic conditions, not merely the frequency of bus dispatches. If Avis dispatches buses every 0.5 minutes, there will be more buses in operation, but the average round trip time remains the same at 20 minutes, assuming no changes in traffic conditions or route speeds.

Answer:

[tex] 28x^2 [/tex]

Step-by-step explanation:

[tex]area \:o f \: \triangle = \frac{1}{2} \times 8x \times 7x \\ = 4x \times 7x \\ = 28 {x}^{2} [/tex]

Answer:

-b/a and line is falling

Step-by-step explanation:

Let (-a,0) and (0,-b) be point 1 and 2, respectively. The slope passing through point (-a,0) and (0,-b) can be calculated using the following equation

[tex]m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-b - 0}{0 - (-a)} = -b/a[/tex]

As b and a are positive real number, then the slope m = -b/a is a negative real number. The line passing through these 2 points is falling.

The slope of the line passing through points (-a, 0) and (0, -b) is calculated using the formula for slope, resulting in a slope of -b/a, which indicates that the line falls as it moves from left to right.

To calculate the slope (m), we use the formula m = (y2 - y1) / (x2 - x1), with (x1, y1) = (-a, 0) and (x2, y2) = (0, -b). Substituting the values, we get m = (-b - 0) / (0 - (-a)) = -b/a. Since both a and b are positive real numbers, the sign of the slope depends on the negative numerator, indicating the line falls as it goes from left to right. If a and/or b were zero, revealing a vertical or horizontal line, a different methodology would apply, such as indicating an undefined slope for a vertical line or a zero slope for a horizontal line.

The function f(x) = x3 - 9x2 - 21x + 8 increases in the intervals (-∞, -1) and (7, ∞), and decreases in the interval (-1, 7). It achieves a local minimum at -190 and a local maximum at -23. The inflection point is at x = 3, with the function being concave up for x < 3 and concave down for x > 3.

Given the function f(x) = x3 - 9x2 - 21x + 8, we need to find intervals where the function is increasing or decreasing, and determine its local minima/maxima, inflection points, and where it is concave up or down.

To find the intervals, we need to first take the derivative of f(x): f'(x) = 3x2 - 18x - 21 which becomes 3(x - 7)(x + 1). The critical points are then 7 and -1.

The function increases in the interval (-∞, -1) and decreases in the interval (-1, 7). The function again increases in the interval (7, ∞).

Local minima and maxima can be found by evaluating the function at the critical points. f(-1) = -23 and f(7) = -190, so the local maximum value of f is -23 and local minimum is -190.

To find the inflection point, we compute the second derivative f''(x) = 6x - 18, which simplifies to 6(x - 3), giving us an inflection point at x = 3.

The function is concave up for x < 3 and concave down for x > 3.

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a) f(x) is increasing for x in (- ∞, - 1) cup(7, infty)

f(x) is decreasing for x \in (- 1, 7)

b)Local minimum=-237 Local maximum=19

c)Point of inflection: (x, y) = (3, - 109)

d)Interval for concave upward curve: x \in (3, ∞)

e)Interval for concave upward curve: x \in (- ∞, 3)

The given function is [tex]f(x) = x ^ 3 - 9x ^ 2 - 21x + 8[/tex]

[tex]f' * (x) = 3x ^ 2 - 18x - 21[/tex]

f^ prime prime (x) = 6x - 18

a)f(x) is increasing for

f' * (x) > 0

[tex]3x ^ 2 - 18x - 21 > 0[/tex]

[tex]x ^ 2 - 6x - 7 > 0[/tex]

[tex]x ^ 2 - 7x + x - 7 > 0[/tex]

x(x - 7) + 1(x - 7) > 0

(x - 7)(x + 1) > 0

x > 7 and x > - 1or x < 7 and x < - 1

f(x) is decreasing for

f' * (x) < 0

3x ^ 2 - 18x - 21 < 0

x ^ 2 - 6x - 7 < 0

x ^ 2 - 7x + x - 7 < 0

x(x - 7) + 1(x - 7) < 0

(x - 7)(x + 1) < 0

x > 7 and x < - 1 or x < 7 and x > - 1

x=(-1,7)

b)for local extrema

f' * (x) = 0

3x ^ 2 - 18x - 21 = 0

x ^ 2 - 6x - 7 < 0

x ^ 2 - 7x + x - 7 = 0

x(x - 7) + 1(x - 7) = 0

(x - 7)(x + 1) = 0

x = 7 and x = - 1

At x=-1:

f(- 1) = (- 1) ^ 3 - 9 * (- 1) ^ 2 - 21(- 1) + 8 = 19

At x=7:

f(7) = (7) ^ 3 - 9 * (7) ^ 2 - 21(7) + 8 = - 237

Local minimum=-237

Local maximum=19

c) For point of inflection:

f^ prime prime (x) = 0

Rightarrow 6x - 18 = 0

Rightarrow x = 3

y = f(3) = (3) ^ 3 - 9 * (3) ^ 2 - 21(3) + 8 = - 109

(x, y) = (3, - 109)

d)Interval for concave upward curve:

f^ prime prime (x) > 0

6x - 18 > 0

x \in (3, ∞)

e)Interval for concave upward curve:

f^ prime prime (x) < 0

6x - 18 < 0

X€(−0,3)

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The probability that all the answers are the same in the section of the exam is 1/8 or 12.5%.

To find the probability that all the answers are the same, we need to consider the number of ways in which all the answers can be the same, divided by the total number of possible outcomes. Since there are only two possible answers (True or False) for each question, there are a total of 2^4 = 16 possible outcomes.

Out of these 16 outcomes, there are only two ways in which all the answers can be the same: either all True or all False.

Therefore, the probability that all the answers are the same is 2/16 = 1/8 = 0.125, or 12.5%.

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The differential equations for x and y are:

x = (cost - D^2 (t2 + 3t)) / ((D+5) + D^2 (D+9))

y = (t2 + 3t - D2 cost) / (D2 / (D+5) + (D+9))

Here's how to solve the problem:

(a) Write the system as a differential equation for x:

Eliminate y: Substitute the second equation into the first equation to eliminate y.

(D+5)x + D2(t2 + 3t) = cost

Rearrange and define operator: Define P(D) = (D+5) + D^2 (D+9).

P(D) x = cost - D^2 (t2 + 3t)

Write as differential equation:

x = (cost - D^2 (t2 + 3t)) / P(D)

(b) Write the system as a differential equation for y:

Eliminate x: Substitute the first equation into the second equation to eliminate x.

D2((cost - D2y) / (D+5)) + (D+9)y = t2 + 3t

Rearrange and define operator: Define Q(D) = D2 / (D+5) + (D+9).

Q(D) y = t2 + 3t - D2 cost

Write as differential equation:

y = (t2 + 3t - D2 cost) / Q(D)

Therefore, the differential equations for x and y are:

x = (cost - D^2 (t2 + 3t)) / ((D+5) + D^2 (D+9))

y = (t2 + 3t - D2 cost) / (D2 / (D+5) + (D+9))

(a)[tex]\(x = \frac{1}{D^3 + 14D + 5}\cdot \left( \cos(t) \right)\)[/tex]

(b)[tex]\(y = \frac{1}{D^3 + 14D + 5}\cdot \left( t^2 + 3t \right)\)[/tex]   by expressing (x) and (y) in terms of differential operators and given functions of (t), we derive their differential equations .

The given system of equations can be transformed into differential equations using operator (D), representing differentiation with respect to a variable, often time.

(a) To derive the differential equation for (x), we isolate (x) in terms of the differential operator \(D\). Rearranging the first equation gives us [tex]\(x = \frac{\cos(t)}{D+5} - \frac{D^2y}{D+5}\).[/tex]Substituting the expression for (y) from the second equation into this yields[tex]\(x = \frac{\cos(t)}{D+5} - \frac{D^2}{D+5}\left(\frac{t^2 + 3t}{D+9}\right)\).[/tex] Simplifying further leads to [tex]\(x = \frac{1}{D^3 + 14D + 5}\cdot \left( \cos(t) \right)[/tex]\).

(b) Similarly, isolating \(y\) from the given equations leads to [tex]\(y = \frac{t^2 + 3t}{D+9} - \frac{D+5}{D+9}x\).[/tex]Substituting the expression for (x) from the first equation into this yields [tex]\(y = \frac{t^2 + 3t}{D+9} - \frac{D+5}{D+9}\left(\frac{\cos(t)}{D+5} - \frac{D^2y}{D+5}\right)\).[/tex]Further simplification results in [tex]\(y = \frac{1}{D^3 + 14D + 5}\cdot \left( t^2 + 3t \right)\).[/tex]

Therefore, by expressing (x) and (y) in terms of differential operators and given functions of (t), we derive their differential equations as specified in the final answer.

Answer:

Step-by-step explanation:

Hello!

X: number of gypsy moths in a randomly selected trap.

This variable is strongly right-skewed. with a standard deviation of 1.4 moths/trap.

The mean number is 1.2 moths/trap, but several have more.

a.

The population is the number of moths found in traps places by the agriculture departments.

The population mean μ= 1.2 moths per trap

The population standard deviation δ= 1.4 moths per trap

b.

There was a random sample of 60 traps,

The sample mean obtained is X[bar]= 1

And the sample standard deviation is S= 2.4

c.

As the text says, this variable is strongly right-skewed, if it is so, then you would expect that the data obtained from the population will also be right-skewed.

d. and e.

Because you have a sample size of 60, you can apply the Central Limit Theorem and approximate the distribution of the sampling mean to normal:

X[bar]≈N(μ;σ²/n)

The mean of the distribution is μ= 1.2

And the standard deviation is σ/√n= 1.4/50= 0.028

f. and g.

Normally the distribution of the sample mean has the same shape of the distribution of the original study variable. If the sample size is large enough, as a rule, a sample of size greater than or equal to 30 is considered sufficient you can apply the theorem and approximate the distribution of the sample mean to normal.

You have a sample size of n=10 so it is most likely that the sample mean will have a right-skewed distribution as the study variable. The sample size is too small to use the Central Limit Theorem, that is why you cannot use the Z table to calculate the asked probability.

I hope it helps!

The ODE is separable:

[tex]\dfrac{\mathrm dV}{\mathrm dt}=2\sqrt V\implies\dfrac{\mathrm dV}{2\sqrt V}=\mathrm dt[/tex]

Integrate both sides to get

[tex]\sqrt V=t+C[/tex]

Assuming the bubble starts with zero volume, so that [tex]V(0)=0[/tex], we find

[tex]\sqrt0=0+C\implies C=0[/tex]

Then the volume of the bubble at time [tex]t[/tex] is

[tex]V(t)=t^2[/tex]

and so the time it take for the bubble to reach a volume of 629 cubic cm is

[tex]t^2=529\implies t=23[/tex]

or 23 seconds after the teen first starts blowing the bubble.

Answer:

r(t) = (e^t +1, -cos(t) + 3, tan(t) + 2)

Step-by-step explanation:

A primitive of e^t is e^t+c, since r(0) has 2 in its first cooridnate, then

e^0+c = 2

1+c = 2

c = 1

Thus, the first coordinate of r(t) is e^t + 1.

A primitive of sin(t) is -cos(t) + c (remember that the derivate of cos(t) is -sin(t)). SInce r(0) in its second coordinate is 2, then

-cos(0)+c = 2

-1+c = 2

c = 3

Therefore, in the second coordinate r(t) is equal to -cos(t)+3.

Now, lets see the last coordinate.

A primitive of sec²(t) is tan(t)+c (you can check this by derivating tan(t) = sin(t)/cos(t) using the divition rule and the property that cos²(t)+sin²(t) = 1 for all t). Since in its third coordinate r(0) is also 2, then we have that

2 = tan(0)+c = sin(0)/cos(0) + c = 0/1 + c = 0

Thus, c = 2

As a consecuence, the third coordinate of r(t) is tan(t) + 2.

As a result, r(t) = (e^t +1, -cos(t) + 3, tan(t) + 2).

To find the function r(t) that satisfies the given condition, integrate each component of r'(t) and solve for the constants using the initial condition.

To find the function r(t) that satisfies the condition r'(t) = (e^t, sin t, sec^2 t) and r(0) = (2, 2, 2), we need to integrate each component of r'(t) with respect to t.

Combining these integrals, we get r(t) = (e^t + C_1, -cos t + C_2, tan t + C_3). Substituting the initial condition r(0) = (2, 2, 2), we can solve for the constants and obtain the function r(t) = (e^t + 2, -cos t + 2, tan t + 2).

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Answer:

405150

Step-by-step explanation:

3 people are to be selected from 75 people. This could be done using the combination function, [tex]\binom{75}{3}[/tex]. But thus function only gives the number of ways of choosing any 3 out of the 75 customers. For any single selection of any 3 customers, there are 3! ways of sharing the three prizes among them. Hence the total number of ways of sharing the prizes is

[tex]\binom{75}{3}\times3! = 405150[/tex]

In fact, this is the permutation function given by

[tex]{}^{75}P_3 = \dfrac{75!}{(75-3)! = 405150}[/tex]

Answer:

a)

[tex]k = \dfrac{1}{21}[/tex]

b) 0.476

c) 0.667    

Step-by-step explanation:

We are given the following in the question:

Y = the number of forms required of the next applicant.

Y: 1, 2, 3, 4, 5, 6

The probability is given by:

[tex]P(y) = ky[/tex]

a) Property of discrete probability distribution:

[tex]\displaystyle\sum P(y_i) = 1\\\\\Rightarrow k(1+2+3+4+5+6) = 1\\\\\Rightarrow k(21) = 1\\\\\Rightarrow k = \dfrac{1}{21}[/tex]

b) at most four forms are required

[tex]P(y \leq 4) = \displaystyle\sum^{y=4}_{y=1}P(y_i)\\\\P(y \leq 4) = \dfrac{1}{21}(1+2+3+4) = \dfrac{10}{21} = 0.476[/tex]

c) probability that between two and five forms (inclusive) are required

[tex]P(2\leq y \leq 5) = \displaystyle\sum^{y=5}_{y=2}P(y_i)\\\\P(2\leq y \leq 5) = \dfrac{1}{21}(2+3+4+5) = \dfrac{14}{21} = 0.667[/tex]

The constant of proportionality (k) equals 1/21. The probability that at most four forms are required is 10/21. The probability that between two and five forms (inclusive) are required is 14/21 or 2/3 when simplified.

The student has provided information regarding a probability distribution which indicates that the probability (p(y)) is proportional to the number of forms required (y), with y ranging from 1 to 6. Since we know probabilities must sum to 1, this allows us to find the constant of proportionality (k).

Part (a): Finding the Value of k

We can use the formula sum of all probabilities equals 1: 1 = k * (1 + 2 + 3 + 4 + 5 + 6). The sum of the numbers from 1 to 6 is 21, therefore 1 = 21k. Solving for k gives us k = 1/21.

Part (b): Probability of At Most Four Forms Required

To find the probability of at most four forms being required, we need to sum the probabilities for 1, 2, 3 and 4 forms: P(x ≤ 4) = k + 2k + 3k + 4k. Substituting the value of k, we get P(x ≤ 4) = (1/21)(1 + 2 + 3 + 4) = 10/21.

Part (c): Probability Between Two and Five Forms Required (Inclusive)

We calculate this by adding the probabilities for 2, 3, 4, and 5 forms: P(2 ≤ x ≤ 5) = 2k + 3k + 4k + 5k. With k = 1/21, this probability is P(2 ≤ x ≤ 5) = (1/21)(2 + 3 + 4 + 5) = 14/21 or 2/3 when simplified.

Answer:

Step-by-step explanation:

We would apply the formula for determining compound interest which is expressed as

y = P(1 + r/n)^nt

Where

y = the value of the investment at the end of t years

r represents the interest rate.

n represents the periodic interval at which it was compounded.

P represents the principal or initial amount invested

From the information given,

P = $4700

r = 4.75% = 4.75/100 = 0.0475

n = 1 because it was compounded once in a year.

Therefore, the exponential function showing the relationship between y and t is

y = 4700(1 + 0.0475/1)^1 × t

y = 4700(1.0475)^t

Answer:

Total member = a+b+c+d+e+f+g=5+4+2+0+5+0+4=20 member

Step-by-step explanation:

Let suppose the attached histogram

a) Total members whose weight 110 = 5

b) Total members whose weight 120 = 4

c) Total members whose weight 130 = 2

d) Total members whose weight 140 = 0

e) Total members whose weight 150 = 5

f) Total members whose weight 160 = 0

g) Total members whose weight 170 = 4

Total member = a+b+c+d+e+f+g=5+4+2+0+5+0+4=20 member

A histogram is a statistical graphical representation of data with the help of bars with different heights.

Answer:

The Manufacturing cycle​ efficiency = 69 %

Step-by-step explanation:

The waiting time is  = 72 minute

The manufacturing lead time is ( [tex]T_{lead}[/tex] )= 160 minutes

Total cycle time ( [tex]T_{c}[/tex] ) = 160 + 72 = 232 minutes

Manufacturing cycle​ efficiency =  [tex]\frac{T_{lead} }{T_{c} }[/tex]

⇒ Manufacturing cycle​ efficiency = [tex]\frac{160}{232}[/tex]

⇒ Manufacturing cycle​ efficiency = 69 %

Therefore, the Manufacturing cycle​ efficiency = 69 %

Answer:

$1 USD = $1.20 Canadian dollars

Step-by-step explanation:

Use proportions of Canadian dollars over US dollars so you get

300 CAD/250 USD = x CAD / 1 USD

solve for x to get 1.2

So $1 USD is equal to $1.20 Canadian dollars.

Answer:

$1.20

Step-by-step explanation:check work by any of these ways.

300/1.20=250,

300/250=1.20,

Or 1.20*250=300