A disease is infecting a colony of 1000 penguins living on a remote island. Let P(t) be the number of sick penguins t days after the outbreak. Suppose that 50 penguins had the disease initially, and suppose that the disease is spreading at a rate proportional to the product of the time elapsed and the number of penguins who do not have the disease.

(a) Give the mathematical model(differential equation and initial condition) for P.

(b) Find the generalsolution of the differential equation in (a).

(c) Find the particular solution that satisfies the initial condition.

Answer:

P(O|R)

Step-by-step explanation:

The conditional probability notation of two events A and B can be written as either P(A|B) or P(B|A).

The '|' sign is read as 'given'. So, P(A|B) is read as the probability of event A given event B which implies that it is the probability that event A will occur given that event B has already occurred.

In the question,

Event R = Person lives in the city of Raleigh

Event O = Person is over 50 years old

The statement says, 'given that the person lives in Raleigh' which means that event R has already occurred and we need to find the probability of event O (the randomly chosen person is over 50 years old).

Hence, this statement can be given in conditional probability notation as

P(O|R)

The question pertains to conditional probability (P(O|R)) and requires specific data to calculate the probability that a randomly chosen person is over 50 years old given they live in Raleigh.

The student is asking about the concept of conditional probability, which in this case refers to the probability of a randomly chosen person being over 50 years old, given that they live in Raleigh. To express this mathematically, we say P(O|R), which means the probability of event O occurring given that event R has occurred.

To find this probability, one would typically use information from a study or a survey giving population counts or percentages of those over 50 within the city of Raleigh population. Without specific data, one cannot provide a precise probability value.

However, the concept is important in probability theory and widely applied across different domains.

Answer:

the probability that Ron will get hit by a pitch exactly once is 36.71%

Step-by-step explanation:

The random variable X= number of times Ron is hits by pitches in 23 plate appearances  follows ,a binomial distribution. Where

P(X=x) = n!/(x!*(n-x)!)*p^x*(1-p)^x

where

n= plate appearances =23

p= probability of being hit by pitches = 21/602

x= number of successes=1

then replacing values

P(X=1) = 0.3671 (36.71%)

The probability that Ron will get hit by a pitch exactly once in his 23 playoff plate appearances, given his regular season hit rate, is approximately 0.37 or 37%.

The subject of this problem is probability; it's asking us to calculate the chances of a specific event happening. It is given that during the regular season, Ron was hit by pitches 21 times out of 602 plate appearances. Thus, the probability of him getting hit by a pitch is 21/602, or approximately 0.035.

In the playoffs, he has 23 plate appearances. We want to find the probability that he gets hit exactly once. This is a binomial probability problem, using the formula:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

Where n is the number of trials (plate appearances), k is the number of successes we want (getting hit by the pitch), p is the success probability, and C(n, k) is the combination operator. Substituting the given values:

P(X=1) = C(23, 1) * (0.035^1) * ((1-0.035)^(23-1))

Performing this calculation gives a pitch hitting probability of about 0.37 or 37%, which means Ron is likely to be hit by one pitch during the 23 plate appearances in the playoffs.

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Answer:

The sample space = {SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF}

Step-by-step explanation:

When we say sample space, we mean the list of all possible outcome from an event. For this even of sales representative presenting at homes., only two outcome is possible. Whether:

1.  the home(s) buys his product (S)

2. the home(s) did not buy his product (F).

Thus from three (3) homes, that will be:

==> [tex]2^{3}[/tex] = 2*2*2 = 8 possible outcomes.

The sample space for the experiment where a sales representative makes presentations at three homes with each home resulting in either a sale (S) or no sale (F) consists of 8 possible outcomes: SSS, SSF, SFS, SFF, FSS, FSF, FFS, and FFF.

To find the sample space for the experiment where a sales representative makes presentations about a product in three homes per day, and in each home, there may be a sale (denoted by S) or there may be no sale (denoted by F), we have to consider all the possible outcomes. Each home has two possible outcomes, meaning that the total sample space consists of 23 = 8 possible combinations for the three presentations.

The sample space S can be written as:

Each combination represents one possible outcome for the day's sales presentations.

Final answer:

To express Δy in terms of Δx, we can use the concept of slope. The formula that expresses Δy in terms of Δx is Δy = (2.5 - y1) / (1.5 - x1) * Δx.

Explanation:

To express Δy in terms of Δx, we can use the concept of slope. The formula for slope is:

Slope = (Δy) / (Δx)

To find the slope between two points, we can use the formula:

Slope = (y2 - y1) / (x2 - x1)

In this case, if y = 2.5 when x = 1.5, we can substitute these values into the formula and simplify:

Slope = (2.5 - y1) / (1.5 - x1)

Since we are only interested in expressing Δy in terms of Δx, we can solve for Δy:

Δy = Slope * Δx

Therefore, the formula that expresses Δy in terms of Δx is:

Δy = (2.5 - y1) / (1.5 - x1) * Δx

If we have y = mx + b and we only know one point (1.5, 2.5), we need a second point or more context for an exact equation. The change in y, Δy, with respect to a change in x, Δx, is found using: Δy = m * Δx.

To express Δy in terms of Δx, you can use the concept of derivatives and the definition of a linear function. Here's a step-by-step solution:

Given information: We know that "y = 2.5" when "x = 1.5."

Setting up the function: Let's assume that the relationship between y and x is linear. In a linear function, the rate of change of y with respect to x is constant. We can write the linear equation in the form: y = mx + b, where m is the slope and b is the y-intercept.

Finding the slope (m): Since linear functions have a constant slope, we need to calculate m. If we assume that y changes by some amount Δy when x changes by Δx, then the slope (m) can be represented as: m = Δy / Δx.

Using the derivative: For a linear equation, dy/dx = m. Therefore, Δy = m * Δx.

Given the specific solution: In the problem, we were given a point (x, y) = (1.5, 2.5). However, we need another point or more information to determine the exact form of the function y in terms of x. Without additional information, we cannot definitively determine the slope.

Assuming a direct variation: In simple cases, we might assume a direct variation (y = kx), but this requires more context. Based on the provided hint, if we use the ratio y/x = k, we can set up an initial formula to start with.

Answer: area = 5,456

Answer:

5456 sq. inches

Step-by-step explanation:

Let width = w,

Then length = w + 26

Perimeter = 2[length + width]

2[(w +26) + w] = 2[2w + 26] = 4w + 52

Perimeter given is 300

So, 4w + 52 = 300

4w = 248

w = 248/4 =62

length = 62 + 26 = 88

Area = length × width

Area = 88 × 62 = 5456 sq. inches

Answer:

a. The price that should be charged if the demand is 40 million gallons is $20.25.

b. The demand decreases by 16 millions of gallons.

Step-by-step explanation:

We know that the price-demand equation for gasoline is given by

[tex]0.1 x + 4 p = 85[/tex]

where

p is the price per gallon in dollars and

x is the daily demand measured in millions of gallons.

a. To find what price should be charged if the demand is 40 million gallons you must

Solve for p,

[tex]0.1x\cdot \:10+4p\cdot \:10=85\cdot \:10\\x+40p=850\\40p=850-x\\p=\frac{850-x}{40}[/tex]

We know that the demand is 40 million gallons (x = 40). So,

[tex]p=\frac{850-40}{40}=\frac{81}{4}=20.25[/tex]

b. To find how much does the demand decrease when the price increases by $0.4 you must

Solve for x,

[tex]0.1x\cdot \:10+4p\cdot \:10=85\cdot \:10\\x+40p=850\\x=850-40p[/tex]

We know that the price increases by $0.4. So,

[tex]-40\left(0.4\right)=-16[/tex]

The demand decreases by 16 millions of gallons.

When the demand is 40 million gallons, the price per gallon should be $20.25. The impact of a $0.4 price increase on the demand can be calculated by substitifying p in the equation, solving for x, and subtracting the original x value.

The subject of this question is algebra, specifically dealing with the use of equations representing real-world scenarios. In this case, the equation represents price-demand dynamics for gasoline.

a. To find the price that should be charged when the demand is 40 million gallons, substitute x with 40 in the equation, which gives 0.1 * 40 + 4p = 85. By simplifying this, we get 4 + 4p = 85. Further solving for p, we get 4p = 81, therefore p = 81 / 4, which is $20.25 per gallon.

b. When the price increases by $0.4, substitute p with p + 0.4 in the equation. This gives 0.1x + 4(p + 0.4) = 85. Solving this for x, and then subtracting the original x value, gives us the decrease in demand due to the increase in price.

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Answer:

The answer to your question is below

Step-by-step explanation:

Inequality 1

                        7(x + 2) - 8 ≥ 13

                        7x + 14 - 8 ≥ 13

                        7x + 6 ≥ 13

                         7x ≥ 13 - 6

                         7x ≥ 7

                           x ≥ 7/7

                           x ≥ 1

Inequality 2

                       8x - 3 < 4x - 3

                        8x - 4x < - 3 + 3

                               4x < 0

                                 x < 0 / 4

                                 x < 0

Interval notation   (-∞ , 0) U [1, ∞)

See the graph below

Answer:

There is a 50% probability that its total life will exceed 20,000 miles.

Step-by-step explanation:

To solve this question, we use the following formula:

[tex]P(A|B) = \frac{P(A \cap B)}{P(B)}[/tex]

In which P(A|B) is the probability of A happening, given that B has happened, [tex]P(A \cap B)[/tex] is the probability of A and B happening, and P(B) is the probability of B happening.

In this problem, we want:

The probability of the total life of the car battery exceeding 20,000 miles, given that it exceeded 10,000 miles.

[tex]P(A \cap B)[/tex] is the probability of exceeding 20,000 and 10,000 miles. It is the same as the probability of exceeding 20,000 miles(If it exceeded 20,000 miles, necessarily it will have exceeded 10,000 miles). So [tex]P(A \cap B) = 0.4[/tex]

P(B) is the probability of exceeding 10,000 miles. So [tex]P(B) = 0.8)[/tex]

So

[tex]P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.4}{0.8} = 0.5[/tex]

There is a 50% probability that its total life will exceed 20,000 miles.

Final answer:

If a new car battery is still working after 10,000 miles, the probability that its total life will exceed 20,000 miles is 0.5 or 50%.

Explanation:

The question pertains to conditional probability, which is the probability of an event occurring given that another event has already occurred. Here, we are asked to find the probability that a new car battery will exceed 20,000 miles given that it has already functioned for more than 10,000 miles. This question essentially requires us to calculate conditional probability.

Given:

To find the conditional probability that its total life will exceed 20,000 miles given it has already worked for over 10,000 miles (P(B|A)), we use the formula:

P(B|A) = P(B & A) / P(A)

However, since any battery that has functioned for more than 20,000 miles must have also functioned for more than 10,000 miles, P(B & A) = P(B), hence:

P(B|A) = 0.4 / 0.8 = 0.5

Therefore, if a new car battery is still working after 10,000 miles, the probability that its total life will exceed 20,000 miles is 0.5 or 50%.

Answer:

Let's make a couple of assumptions to clarify the situation.  First, the coin flipping is fair, that is, each flip is independent of all the others and for each flip, the probabilities of heads and tails are both 1/2.  Second, you have enough money to pay me no matter how many tails are flipped before the first head.

Under those assumptions, the expected amount of money I will win in infinite.  

In decision theory, utility is often used to make decisions rather than money. If my utility is proportional to expected monitory payoff, I should pay whatever I can scrape up, my total assets.  For some reason, economists often assume utility functions have deminishing returns, and eventually flatten out.

In that case, the expected amount of utility payoff will be lower than the maximum utility.  What does that mean for this game?  It means it won't matter to me whether I get some large quantity of money like a trillion dollars, or any larger quantity of money, like a quadrillion dollars.  All my needs are met by a trillion dollars.  That's 240 dollars.  So I certainly shouldn't pay more than $40 to play the game.  As the utility function starts to flatten out earlier, perhaps $30 would come out to be a fair payment.

Answer:

If k = −1 then the system has no solutions.

If k = 2 then the system has infinitely many solutions.

The system cannot have unique solution.

Step-by-step explanation:

We have the following system of equations

[tex]x - 2y +3z = 2\\x + y + z = k\\2x - y + 4z = k^2[/tex]

The augmented matrix is

[tex]\left[\begin{array}{cccc}1&-2&3&2\\1&1&1&k\\2&-1&4&k^2\end{array}\right][/tex]

The reduction of this matrix to row-echelon form is outlined below.

[tex]R_2\rightarrow R_2-R_1[/tex]

[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\2&-1&4&k^2\end{array}\right][/tex]

[tex]R_3\rightarrow R_3-2R_1[/tex]

[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\0&3&-2&k^2-4\end{array}\right][/tex]

[tex]R_3\rightarrow R_3-R_2[/tex]

[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\0&0&0&k^2-k-2\end{array}\right][/tex]

The last row determines, if there are solutions or not. To be consistent, we must have k such that

[tex]k^2-k-2=0[/tex]

[tex]\left(k+1\right)\left(k-2\right)=0\\k=-1,\:k=2[/tex]

Case k = −1:

[tex]\left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&-1-2\\0&0&0&(-1)^2-(-1)-2\end{array}\right] \rightarrow \left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&-3\\0&0&0&-2\end{array}\right][/tex]

If k = −1 then the last equation becomes 0 = −2 which is impossible.Therefore, the system has no solutions.

Case k = 2:

[tex]\left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&2-2\\0&0&0&(2)^2-(2)-2\end{array}\right] \rightarrow \left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&0\\0&0&0&0\end{array}\right][/tex]

This gives the infinite many solution.

We use matrix row reduction to determine the values of k that result in no solution, a unique solution, or infinitely many solutions in the system of equations.

To determine the values of k for which the system of equations has no solution, a unique solution, or infinitely many solutions, we will use the concept of matrix row reduction. First, let's rewrite the system of equations in augmented matrix form:

[1 -2 3 2 | 0] [2 1 1 -1 | 0] [2 -1 4 -k^2 | 0]

Performing row reduction on this augmented matrix, we can find the values of k where each situation occurs. If there is a row of 0's followed by a non-zero constant (in the rightmost column), then the system has no solution. If the row reduction yields a matrix with a non-zero row followed by zeroes (except for the last row), then the system has infinitely many solutions. Otherwise, the system has a unique solution.

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Answer:

S ={(1+1=2), (1+2=3), (1+3=4), (1+4=5), (1+5=6), (1+6=7),

     (2+1=3), (2+2=4),(2+3=5),(2+4=6),(2+5=7),(2+6=8),

     (3+1=4), (3+2=5),(3+3=6),(3+4=7),(3+5=8),(3+6=9),

     (4+1=5), (4+2=6),(4+3=7),(4+4=8),(4+5=9),(4+6=10),

     (5+1=6), (5+2=7),(5+3=8),(5+4=9),(5+5=10),(5+6=11),

     (6+1=7), (6+2=8),(6+3=9),(6+4=10),(6+5=11),(6+6=12)}

Step-by-step explanation:

By definition the sample space of an experiment "is the set of all possible outcomes or results of that experiment".

For the case described here: "Toss a six-sided die twice and record the sum of the results".

Assuming that we have a six sided die with possible values {1,2,3,4,5,6}

The sampling space denoted by S and is given by:

S ={(1+1=2), (1+2=3), (1+3=4), (1+4=5), (1+5=6), (1+6=7),

     (2+1=3), (2+2=4),(2+3=5),(2+4=6),(2+5=7),(2+6=8),

     (3+1=4), (3+2=5),(3+3=6),(3+4=7),(3+5=8),(3+6=9),

     (4+1=5), (4+2=6),(4+3=7),(4+4=8),(4+5=9),(4+6=10),

     (5+1=6), (5+2=7),(5+3=8),(5+4=9),(5+5=10),(5+6=11),

     (6+1=7), (6+2=8),(6+3=9),(6+4=10),(6+5=11),(6+6=12)}

The possible values for the sum are 2,3,4,5,6,7,8,9,10,11,12

Answer:

addition tioin multiplication

Step-by-step explanation:

Using the given data, the coefficient of variation is 30% which matches option b.

Complete Question :

The hourly wages of a sample of 130 system analysts are given below. mean = 60 range = 20 mode = 73 variance = 324 median = 74. The coefficient of variation equals a. 0.30%. b. 30% O c. 5.4% d. 54%.

Answer:

a) 4.5

b) x = 11.2, s = 4.65

c) 93.33%                                                

Step-by-step explanation:

We are given he following data in the question:

8, 12, 11, 15, 14, 10, 8, 3, 8, 7, 21, 12, 9, 19, 11

a) Estimation of standard deviation using range

Sorted data: 3, 7, 8, 8, 8, 9, 10, 11, 11, 12, 12, 14, 15, 19, 21

Range = Maximum - Minimum = 21 - 3 = 18

Range rule thumb:

[tex]s = \dfrac{\text{Range}}{4} = \dfrac{18}{4} = 4.5[/tex]

b) Mean and standard deviation

Formula:

[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]  

where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.  

[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]

[tex]Mean =\displaystyle\frac{168}{15} = 11.2[/tex]

Sum of squares of differences = 302.4

[tex]S.D = \sqrt{\dfrac{302.4}{14}} = 4.65[/tex]

c)  fraction of the scores actually lie in the interval x ± 2s

[tex]x \pm 2s = 11.2 \pm 2(4.65) = (1.9,20.5)[/tex]

Since 14 out of 15 entries lie in this range, we can calculate the percentage as,

[tex]\dfrac{14}{15}\times 100\% = 93.33\%[/tex]

Final answer:

To find the work needed to pump all of the water to the top of the tank, we need to calculate the potential energy at the half-full level and the top of the tank. By using the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height, we can calculate the potential energy for each level and find the difference. The work needed is approximately 41 million mega-joules.

Explanation:

To calculate the work needed to pump all the water to the top of the tank, we need to find the potential energy of the water at the half-full level and the potential energy of the water at the top of the tank. The potential energy of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.

Given that the tank is half full, the height of the water column is 18/2 = 9 meters. The radius of the tank is 4 meters, so the volume of the water is πr^2h = π(4^2)(9) = 144π cubic meters. The mass of the water is density × volume = 1000 × 144π = 144,000π kg.

The potential energy at the half-full level is PE = mgh = 144,000π × 9 × 9.8 = 12,931,200π joules. The potential energy at the top of the tank is PE = mgh = 144,000π × 18 × 9.8 = 25,862,400π joules. Therefore, the work needed to pump all the water to the top of the tank is the difference in potential energy = 25,862,400π - 12,931,200π = 12,931,200π joules. To convert to mega-joules, divide by 1,000,000, so the work needed is approximately 41 million mega-joules.

Answer:

(9 weeks 5 days) - (1 week 6 days) =  55 days

Step-by-step explanation:

Answer:

Step-by-step explanation:

Answer:

The variance is 2.5.

Step-by-step explanation:

Let X = number of visitors in a museum.

The random variable X has an average of  5 visitors per 2 minutes.

Then in 1 minute the average number of visitors is, [tex]\frac{5}{2} =2.5[/tex]

The random variable X follows a Poisson distribution with parameter λ = 2.5.

The variance of a Poisson distribution is:

[tex]Variance=\lambda[/tex]

The variance of this distribution is:

[tex]V(X)=\lambda=2.5[/tex]

Thus, the variance is 2.5.

Answer:

The question is lacking the image of the histogram, but the file attachment to this answer contains the complete question and histogram image.

The percentage of the rats that navigated the maze in less than 5.5 minutes is 84%

Step-by-step explanation:

First of all let us compute the total frequency which represents the total number of rats in the experiment.

from the information given in the question, we are told that the total number of rats involved in the experiment are 25 rats and this makes up the total frequency.

To calculate the percentage of rats that navigated the maze in less than 5.5 minutes, you will first of all need to identify the 5.5 minute point on the histogram and add all the frequencies below that point. This gives the total number of rat that navigated in less than 5.5 minutes. These frequencies from 0 to <5.5 are; 3, 8, 6, and 4. And the total is given as the sum of these frequencies shown below:

Total number of rats that navigated in less than 5.5 minutes = 3 + 8 + 6 + 4 = 21. Hence, a total of 21 rats navigated the maze in less than 5.5 minutes.

Now to find the percentage of the number of rats that navigated in less than 5.5 minutes, we have to find out what percentage of 25 (total number of rats) is 21 (number of rats that navigated in less than 5.5 minutes). This is calculated thus:

[tex]\frac{21}{25}[/tex] × 100 = 0.84 × 100 = 84 %.

Therefore 84% of the total number of rats navigated the maze in less than 5.5 minutes in the experiment.

To find the percentage of rats that navigated the maze in less than 5.5 minutes, locate the bar on the histogram that represents that interval and calculate the percentage based on the number of rats in that interval compared to the total number of rats.

The histogram shows the times required for 25 rats to navigate a maze. To find the percentage of rats that navigated the maze in less than 5.5 minutes, we need to look at the data on the histogram. The histogram is divided into various time intervals, and the bars represent the number of rats that fall into each interval. You need to locate the bar that corresponds to the interval less than 5.5 minutes and calculate the percentage of rats represented by that bar.

Let's say the bar for the interval less than 5.5 minutes represents 10 rats. To calculate the percentage, we divide the number of rats in that interval (10) by the total number of rats (25) and multiply by 100:

(10/25) x 100 = 40%

Therefore, 40% of the rats navigated the maze in less than 5.5 minutes.

Answer:

a) 50,000 bottles

b) x = 45

Step-by-step explanation:

We are given the following in the question:

The annual revenue of sun block depends on how much they sell.

Let x be the quantity of 5 oz. bottles of sun block that they make and sell each year measured in 1000 's of bottles.

For x = 10,

10,000 bottles were made and sell each year.

For x = 25,

25,000 bottles of sun block were made and sell each year.

a) x = 50

[tex]\text{Number of bottles} = 50\times 1000 = 50,000[/tex]

Thus, 50,000 bottles of sun block does Scura make and sell.

b) Scura produces and sells 45000 bottles of sunblock

We have to find the value of x

[tex]x = \displaystyle\frac{\text{Number of bottles}}{1000} = \frac{45000}{1000} = 45[/tex]

Thus, x = 45 if Scura produces and sells 45000 bottles of sunblock.

For Scura's Sunblock Sales, if x=50, they sell 50,000 bottles, and when Scura sells 45,000 bottles of sunblock, x equals 45.

a. If x=50, then according to the relationship given where x represents thousands of bottles, Scura makes and sells 50,000 bottles of sun block.

b. To determine what x is equal to when Scura produces and sells 45,000 bottles of sunblock, we take the total number of bottles and divide by 1000, since x is measured in 1000s. So, x=45 when Scura produces and sells 45,000 bottles of sunblock.

Answer:

[tex]$ \frac{\textbf{1.55}}{\textbf{1}} \hspace{1mm} \textbf{=} \hspace{1mm} \frac{\textbf{x}}{\textbf{3.5}} $[/tex]

Step-by-step explanation:

Let us assume that the total cost of the grapes = x $.

Given that it costs 1.55 $ for one pound. We are asked to determine how much it would cost for 3.5 pounds.

note that for one pound you pay 1.55 so how much should you pay for 3.5 pounds? Clearly, the cost increases with the increase in weight of the item.

This is equal to [tex]$ \frac{1.55}{1} \times 3.5 = x $[/tex]

[tex]$ \iff \frac{1.55}{1} = \frac{x}{3.5} $[/tex]

Hence, the answer.

The cone [tex]z=\sqrt{x^2+y^2}[/tex] and the sphere [tex]z=\sqrt{4-x^2-y^2}[/tex] intersect in a circle of radius [tex]\sqrt 2[/tex] in the plane [tex]z=\sqrt2[/tex]:

[tex]\sqrt{x^2+y^2}=\sqrt{4-x^2-y^2}\implies 2x^2+2y^2=4\implies x^2+y^2=2[/tex]

[tex]\implies z=\sqrt{x^2+y^2}=\sqrt2[/tex]

In Cartesian coordinates, the volume is then given by the integral

[tex]\displaystyle\int_{-\sqrt2}^{\sqrt2}\int_{-\sqrt{2-x^2}}^{\sqrt{2-x^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{4-x^2-y^2}}\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]

In cylindrical coordinates, the integral is

[tex]\displaystyle\int_0^{2\pi}\int_0^{\sqrt2}\int_r^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta[/tex]

For the given problem, the volume of the structure can be calculated using both Cartesian and cylindrical coordinates. For Cartesian coordinates, the coordinates represents as  [tex]z=x^2+y^2[/tex]and  [tex]x^2+y^2+z^2=4.[/tex]For cylindrical coordinates, equations represent as [tex]z=r^2[/tex]and [tex]r^2+z^2=4[/tex]. Both integrations describe the volume of the structures.

In the given problem, the volume contains two geometric shapes: the cone and the sphere. The volume of the shape can be calculated using both Cartesian and cylindrical coordinates.

Using Cartesian coordinates, first describe cone and sphere as  [tex]z=x^2+y^2[/tex] and [tex]x^2+y^2+z^2=4[/tex] respectively. Define the volume by double integration:

∫∫ D (4 - z) dxdy

Where D is the region in the xy-plane bounded by the projection of the volume.

Using cylindrical coordinates, we represent the figures as [tex]z=r^2[/tex]and  [tex]r^2+z^2=4.[/tex] The volume integral in cylindrical coordinates is then given by:

∫ (from 0 to 2pi) ∫ (from 0 to √2) ∫ (from  [tex]r^2 \ to \ 2-r^2[/tex]) rdzdrdθ

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Answer:

There is no enough evidence to claim that the average cost of accesories is different from $3,000.

Step-by-step explanation:

The significance level for this test is α=0.05.

The classical method is based on regions of rejection of acceptance, according to the sample parameter. In this case, the standard deviation of the population is unknown.

The null and alternative hypothesis are:

[tex]H_0: \mu=3000\\\\ H_a: \mu\neq 3000[/tex]

This is a two-tailed test, with significance level of 0.05.

The t-value for this sample is:

[tex]t=\frac{x-\mu}{s/\sqrt{N}} =\frac{3256-3000}{2300/\sqrt{50}}=\frac{256}{325}=0.787[/tex]

The degrees of freedom are:

[tex]df=n-1=50-1=49[/tex]

For df=49 and α=0.05 (two-tailed test), the critical values are [tex]|t|>2.009[/tex], so the value t=0.787 is within the acceptance region.

The null hypothesis can not be rejected.

Answer:

1.33 s

Step-by-step explanation:

Given:

Δy = -5 ft

v₀ = 17.5 ft/s

a = -32 ft/s²

Find: t

Δy = v₀ t + ½ at²

-5 = 17.5 t + ½ (-32) t²

0 = -16t² + 17.5t + 5

0 = 32t² − 35t − 10

t = [ 35 ± √((-35)² − 4(32)(-10)) ] / 64

t = (35 ± √2505) / 64

t = 1.33

Amy has 1.33 seconds to react before the volleyball hits the ground.

Amy has to react to the volleyball based on its initial upward velocity and the height at which it was hit, using gravitational equations to calculate the time before it hits the ground.

Patricia serves the volleyball to Amy with an upward velocity of 17.5ft/s. The ball is 5 feet above the ground when she strikes it. To determine how long Amy has to react before the volleyball hits the ground, we can use the equations of motion under gravity. Assuming the acceleration due to gravity (g) is 32.2ft/s2 (downward), the time (t) for the volleyball to reach the ground can be found by solving the quadratic equation derived from the formula:

h = v0t - (1/2)gt2

where h is the height above the ground (5 feet), v0 is the initial velocity (17.5ft/s), and t is the time in seconds. One can use the quadratic formula to solve for t. However, to provide a concrete example and simplify the calculation for the purpose of this answer, we would use a calculator or other computational tools to solve for t numerically, remembering to consider the positive root that makes physical sense. One would typically find a time in the range of a second or slightly more for this scenario.

Answer:

b. 0.6592 and 0.7044

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

A survey of an urban university showed that 750 of 1,100 students sampled attended a home football game during the season. This means that [tex]n = 1100, p = \frac{750}{1100} = 0.6818[/tex]

90% confidence interval

So [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.6818 - 1.645\sqrt{\frac{0.6818*0.3182}{1100}} = 0.6592[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.6818 + 1.645\sqrt{\frac{0.6818*0.3182}{1100}} = 0.7044[/tex]

So the correct answer is:

b. 0.6592 and 0.7044

Answers: 0.286

Explanation:

Let E → major in Engineering

Let S → Play club sports

P (E) = 28% = 0.28

P (S) = 18% = 0.18

P (E ∩ S ) = 8% = 0.08

Probability of student plays club sports given majoring in engineering,

P ( S | E ) = P (E ∩ S ) ÷ P (E) = 0.08 ÷ 0.28 = 0.286

To find the probability that a student plays club sports given that they major in engineering, use conditional probability.

To find the probability that a student plays club sports given that they major in engineering, we need to use conditional probability.

The formula for conditional probability is:

P(A|B) = P(A ∩ B) / P(B)

In this case, A represents playing club sports and B represents majoring in engineering. We are given that P(A ∩ B) = 8% and P(B) = 28%.

Plugging these values into the formula, we get:

P(A|B) = 8% / 28% = 0.2857

So the probability that a student plays club sports given that they major in engineering is approximately 0.2857, or 28.57%.

Answer:

For the first case we are going to assume that the order matters, on this case 6, H is not the same as H,6

The sampling space denoted by S and is given by:

S ={(1,H), (2,H),(3,H),(4,H),(5,H),(6,H),

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

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

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

If we consider that (5,H) is equal to (H,5) "order no matter" then we will have just 12 elements in the sampling space:

S ={(1,H), (2,H),(3,H),(4,H),(5,H),(6,H),

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

Step-by-step explanation:

By definition the sample space of an experiment "is the set of all possible outcomes or results of that experiment".

For the case described here: "Toss a coin and a six-sided die".

Assuming that we have a six sided die with possible values {1,2,3,4,5,6}

And for the coin we assume that the possible outcomes are {H,T}

For the first case we are going to assume that the order matters, on this case 6, H is not the same as H,6

The sampling space denoted by S and is given by:

S ={(1,H), (2,H),(3,H),(4,H),(5,H),(6,H),

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

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

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

If we consider that (5,H) is equal to (H,5) "order no matter" then we will have just 12 elements in the sampling space:

S ={(1,H), (2,H),(3,H),(4,H),(5,H),(6,H),

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

To determine if the nutrition label is accurate, a hypothesis test can be conducted using the provided sample data. Calculating the z-score and finding the p-value will determine if there is evidence that the label is inaccurate.

In order to determine if there is evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips, we can conduct a hypothesis test using the given sample data. Let's state the null and alternative hypotheses:

Null Hypothesis (H0): The nutrition label provides an accurate measure of calories in the bags of potato chips.

Alternative Hypothesis (Ha): The nutrition label does not provide an accurate measure of calories in the bags of potato chips.

To test these hypotheses, we can calculate the z-score using the formula:

z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))

In this case, the population mean is 130 calories (as stated on the nutrition label), the sample mean is 134 calories, the population standard deviation is 17 calories (as given), and the sample size is 35 bags (as given). Plugging in these values, we can calculate the z-score.

Once we have the z-score, we can find the p-value associated with it from a standard normal distribution table or using statistical software. If the p-value is less than the chosen significance level (usually 0.05), we reject the null hypothesis and conclude that there is evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips.

Without knowing the calculated p-value, we cannot make a decision based on the given random sample.

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Answer:

a. $1773.82

b. $1592.1

Step-by-step explanation:

1. If he pays k dollar in the first year, then the amount that he owned without interest is

7277 - k

The amount that he owned including interest of 11% in the 2nd year is

(7277 - k)*1.11 or 7277*1.11 - 1.11k

After 2nd year and paying k then the amount he owned (without interest is)

(7277 - k)*1.11 - k

With interest

[(7277 - k)*1.11 - k]1.11 or [tex]7277*1.11^2 - 1.11^2k - 1.11k[/tex]

So after 5 years

[tex]7277*1.11^5 - (1.11^5 + 1.11^4 + 1.11^3 + 1.11^2 +1.11)k[/tex]

[tex]12262.17 - 6.91 k[/tex]

Since he's dept-free after 5 year then

[tex]12262.17 - 6.91 k = 0[/tex]

[tex]k = 12262.17 / 6.91 = 1773.82[/tex] dollar

2. The total amount he would have to pay over 5 years is 5k = 5*1773.82 = 8869.1

So the interest we has to pay over 5 years is the total subtracted by the principal, which is 8869.1 - 7277 = 1592.1 dollar

Answer:

The probability that of the 3 households randomly selected at least 1 owns a sports car is 0.1956.

Step-by-step explanation:

Let X = number of household owns a sports car.

The probability of X is, P (X) = p = 0.07.

Then the random variable X follows a Binomial distribution with n = 3 and p = 0.07.

The probability function of a binomial distribution is:

[tex]P(X=x) = {n\choose x}p^{x}[1-p]^{n-x}\\[/tex]

Compute the probability that of the 3 households randomly selected at least 1 owns a sports car:

[tex]P(X\geq 1)=1-P(X<1)\\=1-P(X=0)\\=1- {3\choose 0}(0.07)^{0}[1-0.07]^{3-0}\\=1-0.8044\\=0.1956[/tex]

Thus, the probability that of the 3 households randomly selected at least 1 owns a sports car is 0.1956.

Answer:

part 1) [tex]13[/tex]

part 2) [tex]\frac{119}{45}[/tex]

part 3) [tex]2[/tex]

part 4) [tex]\frac{1,958,309}{128}[/tex]

part 5) [tex]4\ yd^2[/tex]

Step-by-step explanation:

The complete question in the attached figure

we know that

Applying PEMDAS

P ----> Parentheses first  

E -----> Exponents (Powers and Square Roots, etc.)  

MD ----> Multiplication and Division (left-to-right)  

AS ----> Addition and Subtraction (left-to-right)

Part 1) we have

[tex]\frac{2}{3}(6)+\frac{3}{4}(12)[/tex]

Remember that when multiply a fraction by a whole number, multiply the numerator of the fraction by the whole number and maintain the same denominator

so

[tex]\frac{12}{3}+\frac{36}{4}[/tex]

[tex]4+9=13[/tex]

Part 2) we have

[tex]2\frac{1}{3}(3\frac{2}{5}:3)[/tex]

Convert mixed number to an improper fraction

[tex]2\frac{1}{3}=2+\frac{1}{3}=\frac{2*3+1}{3}=\frac{7}{3}[/tex]

[tex]3\frac{2}{5}=3+\frac{2}{5}=\frac{3*5+2}{5}=\frac{17}{5}[/tex]

substitute

[tex]\frac{7}{3}(\frac{17}{5}:3)[/tex]

Solve the division in the parenthesis (applying PEMDAS)

[tex]\frac{7}{3}(\frac{17}{15})[/tex]

[tex]\frac{119}{45}[/tex]

Part 3) we have

[tex]\frac{7}{8}:(1\frac{1}{4}:4)[/tex]

Convert mixed number to an improper fraction

[tex]1\frac{1}{4}=1+\frac{1}{4}=\frac{1*4+1}{4}=\frac{5}{4}[/tex]

substitute

[tex]\frac{7}{8}:(\frac{5}{4}:4)[/tex]

Solve the division in the parenthesis (applying PEMDAS)

[tex]\frac{7}{8}:(\frac{5}{16})[/tex]

Multiply in cross

[tex]\frac{80}{40}=2[/tex]

Part 4) we have

[tex]18:(\frac{2}{3})^2+25:(\frac{2}{5})^7[/tex]

exponents first

[tex]18:(\frac{4}{9})+25:(\frac{128}{78,125})[/tex]

Solve the division

[tex](\frac{162}{4})+(\frac{1,953,125}{128})[/tex]

Find the LCD

LCD=128

so

[tex]\frac{32*162+1,953,125}{128}[/tex]

[tex]\frac{1,958,309}{128}[/tex]

Part 5) Find the area of triangle

The area of triangle is equal to

[tex]A=\frac{1}{2}(b)(h)[/tex]

substitute the given values

[tex]A=\frac{1}{2}(6)(1\frac{1}{3})[/tex]

Convert mixed number to an improper fraction

[tex]1\frac{1}{3}=1+\frac{1}{3}=\frac{1*3+1}{3}=\frac{4}{3}[/tex]

substitute

[tex]A=\frac{1}{2}(6)(\frac{4}{3})=4\ yd^2[/tex]

Answer:

d. skewed right

Step-by-step explanation:

The shape of the given distribution is rightly skewed. For a symmetric distribution mean and median are equal and if mean is greater than median then the distribution is rightly skewed and if mean is less than median then the distribution is skewed left.

In the given distribution mean is greater than median and so the given distribution is skewed right.