Find the mode for the following data set:33 28 22 11 11 17

The differential equation whose general solution is y = a sin 5x – b cos 5x, where a and b are arbitrary constants, is given by d²y/dx² + 25a y = 0

To obtain the differential equation, we need to take the derivative of y with respect to x

y = a sin 5x – b cos 5x

dy/dx = 5a cos 5x + 5b sin 5x

Now, we can take the second derivative of y with respect to x

d²y/dx² = -25a sin 5x + 25b cos 5x

We can simplify this expression by using the identity sin(θ + π/2) = cos(θ), which implies that cos(θ - π/2) = sin(θ)

d²y/dx² = -25a sin(5x - π/2)

This is the second derivative of y with respect to x. To find the differential equation, we need to express this equation in terms of y and its derivatives. We can use the identity sin(θ - π/2) = -cos(θ) to rewrite the above equation as

d²y/dx² + 25a y = 0

This is the differential equation whose general solution is y = a sin 5x – b cos 5x, where a and b are arbitrary constants

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For an exponential probability distribution of time (in years) until part failure for a certain car, probability that the time until the first critical-part failure is 6 year or more is equals to the 0.181.

The exponential distribution is a type of continuous probability distribution that is used to measure the expected time for an event to occur. Formula is written as [tex]f(X)=\lambda e^{-\lambda x}; X >0, [/tex]

where λ --> rate parameter

X --> observed value

We have time (in year) first critical-part failure for a certain car is exponentially distributed. Let X be the time part failure for a certain car. Now, X follows the exponential distribution with mean 3.5 years. The probability density function of X is [tex]f(X) = \lambda e^{- \lambda x} ;X > 0 [/tex], Here in this problem,

[tex]\lambda = \frac{1}{3.5} [/tex]

= 0.285

Using the formula the probability of

X more than and equal to 6 years is [tex]P( X≥ 6) = 1 - P( X≤6) [/tex]

[tex]= 1- (1 – e^{−0.285×6})[/tex]

[tex]= e^{−0.285×6})[/tex]

= 0.180865 ~ 0.181. Hence, the required probability value is 0.181.

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The height of the cuboid is 8 cm.

What is Volume?

Volume is a measure of the amount of space occupied by a three-dimensional object or substance. It is expressed in cubic units such as cubic meters (m³) or cubic centimeters (cm³).

We first need to convert the width to cm since the length and height are given in cm:

20 mm = 2 cm (since 1 cm = 10 mm)

The formula for the volume of a cuboid is V = lwh, where l is the length, w is the width, and h is the height. We can solve for h by rearranging the formula:

h = V / lw

Substituting the given values, we get:

h = 176 / (11 x 2)

h = 8

Therefore, the height of the cuboid is 8 cm.

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The equation of the line tangent to the cost function at q= 130 is y = -80x + 10550

To find the equation of the line tangent to the cost function at q = 130, we need to calculate the slope of the cost function at that point. This is done by taking the derivative of the cost function with respect to q and evaluating it at q = 130.

The derivative of the cost function is given by: cost'(q) = 50 - q. Evaluating this at q = 130 gives us a slope of -80. Therefore, the equation of the tangent line is given by:

y = mx + b, where m is the slope and b is the y-intercept. Substituting m = -80 and (130, cost(130)) = (130, 4550) as the point on the line, we get:

y = -80x + 10550

This means that at q = 130, the cost of producing one additional gizmo is $80. The y-intercept of 10550 represents the total cost of producing 130 gizmos.

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Answer: a^2 + b^2 = c^2

Step-by-step explanation:

because theres 3 sides of a right triangle

Answer:

30.8 inches.

Step-by-step explanation:

The Pythagorean theorem states that a^2+b^2=c^2, where c is the hypotenuse, or longest side of the triangle opposite to the right angle, and a and b are the two legs of the triangle (the other two sides.)

In the question, we are imagining the TV as a triangle, where its width and height are two sides, and the length diagonally across is the hypotenuse. If its width and height are 25in and18 in, then that would mean that a and b in the equation are 25 and 18. Remember that it doesn't matter which variable you assign (a or b), both are just legs of the triangle.

Plugging 25 and 18 into the equation, we will get that 25^2+18^2=c^2, with c being the diagonal size of Isaac's TV. 25 x 25 = 625 and 18 x 18 = 324, so 625 + 324 = c^2, or, c^2 = 949.

However, this is still c^2, not c, so we must take the square root of both sides. This is because the equation is now essentially saying that c x c = 949, so if we find a number that when multiplied by itself equals 949, that number will be c. to do this, we must take the square root of 949. This will give us an irrational number- approximately 30.8058436. The question asks to round it to the nearest tenth though, so this will simply become 30.8.

In the context of this question, it means that Isaac's television is 30.8 inches long diagonally.

The probability that exactly 8 customers will arrive in a one-hour period is 0.1563, or approximately 15.63%. The probability of exactly 8 customers arriving in a one-hour period can be calculated using the Poisson distribution formula.

The formula is P(X=x) = (e^-λ * λ^x) / x!, where λ is the mean number of customers arriving in a one-hour period and x is the number of customers we want to calculate the probability for.
So in this case, λ = 8.60 and x = 8. Plugging these values into the formula, we get:
P(X=8) = (e^-8.60 * 8.60^8) / 8!
P(X=8) = 0.1563
Therefore, the probability that exactly 8 customers will arrive in a one-hour period is 0.1563, or approximately 15.63%.

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The net profit margin for this shirt is 55%.

What is net profit?

Net profit is the amount of money your business earns after deducting all operating, interest and tax expenses over a period of time. To arrive at this value, you need to know the company's gross profit. If the value of net profit is negative, it is called net loss.

Net profit is another important parameter that determines the financial health of your business. It shows whether the company can earn more than it spends. Net profit helps you decide when and how to expand your business and when to cut costs.

It is important for a business owner to know the difference between profit and profitability. Profit is an absolute number equal to revenues minus expenses. Profitability, on the other hand, is a relative number (percentage) that equals the ratio of profit to revenue.

Profitability is a measure of efficiency and is useful in determining the success or failure of a business.

Given,

Revenue = $12

Cost of goods sold = $3

Operating expenses = 20% of revenue = 20/100 x $12 = $2.4

We know,

Net Profit = Revenue - Cost of goods sold - Operating expenses

[tex]Net \: Profit = 12 - 3 - 2.4[/tex]

Hence, Net Profit = $6.6

Again,

Net Profit Margin = (Net Profit / Revenue) x 100%

Net Profit Margin [tex]= ( \frac{6.6}{ 12}) \times 100\%[/tex]

SO, Net Profit Margin = 55%

Therefore, the net profit margin for this shirt is 55%.

So, it is clear that choice A is correct.

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Solve for y:
y = tan(3x² + C)

To solve the differential equation dy = 6x(y² + 1)dx, follow these steps:

Provided an answer in the form of an(3x³ + K), it seems there may be a typo in the original equation. Please double-check the equation and provide the correct one if necessary.

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The company also believes that winning the hotel bid is independent of winning the office building bid. a. The probability of winning both contracts is 10% .b. The probability of winning at least one contract is 55%. c. The probability of losing both contracts is 45%.

a. The probability the company will win both contracts is the product of the probabilities of winning each bid:

P(winning hotel bid) x P(winning office building bid) = 0.40 x 0.25 = 0.10

So there is a 10% chance the company will win both contracts.

b. The probability the company will win at least one contract is equal to 1 minus the probability of losing both contracts:

P(winning at least one contract) = 1 - P(losing both contracts)

To find the probability of losing both contracts, we can use the complement rule:

P(losing both contracts) = 1 - P(winning hotel bid) x P(winning office building bid)

P(losing both contracts) = 1 - 0.10 = 0.90

Therefore,

P(winning at least one contract) = 1 - 0.90 = 0.10

So there is a 90% chance the company will win at least one contract.

c. The probability the company will lose both contracts is:

P(losing both contracts) = 1 - P(winning hotel bid) x P(winning office building bid)

P(losing both contracts) = 1 - 0.10 = 0.90

So there is a 90% chance the company will lose both contracts.

a. To find the probability of winning both contracts, since they are independent events, simply multiply the probabilities of winning each contract:

P(both) = P(hotel) * P(office) = 0.40 * 0.25 = 0.10, or 10%.

b. To find the probability of winning at least one contract, first calculate the probability of losing both contracts, and then subtract that from 1:

P(at least one) = 1 - P(lose both)

c. To find the probability of losing both contracts, multiply the probabilities of losing each contract:

P(lose hotel) = 1 - P(hotel) = 1 - 0.40 = 0.60
P(lose office) = 1 - P(office) = 1 - 0.25 = 0.75

P(lose both) = P(lose hotel) * P(lose office) = 0.60 * 0.75 = 0.45, or 45%.

Now, calculate the probability of winning at least one contract:

P(at least one) = 1 - P(lose both) = 1 - 0.45 = 0.55, or 55%.

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You can similarly calculate probabilities for other combinations of favorite activities and whether they play a sport or not. Remember to always divide the number of people in the desired category by the total number of people in the table.

It seems that the table you provided is not well-formatted, making it difficult to understand. However, I will do my best to provide a general answer based on the given information.

When using a table with favorite activities and whether a person plays a sport, you can determine the probability of selecting a person with specific characteristics by dividing the number of people with that characteristic by the total number of people in the table.

For example, if you want to find the probability of selecting someone who likes Bowling (B) and plays a sport, you would divide the number of people in that category (78) by the total number of people (200).

P(B and plays a sport) = 78 / 200 = 0.39

You can similarly calculate probabilities for other combinations of favorite activities and whether they play a sport or not. Remember to always divide the number of people in the desired category by the total number of people in the table.

The complete question is-

The following table contains sample data of people who go to Round One which is a multi- entertainment facility that offers Bowling, Arcade Games, Billiards, Karaoke, Ping Pong, Darts, and more. The columns represent their favorite activities, and the rows represent whether they play a sport. A person is selected at random from this group. Use the table to answer the following questions. Total in the rows Bowling (B) 78 30 Karaoke (K) 10 Ping Pong (P-P) 12 6 Darts (D) 5 Plays a sport Does not play a sport Column total 105 95 14 45 108 24 18 50 200 (a) What is the probability that someone prefers Karaoke or plays a sport? Make sure to use proper notation. (Write any formulas used out entirely as part of your work shown) (b) What is the probability that someone plays a sport and prefers Ping-pong? Make sure to use proper notation. (Write any formulas used out entirely as part of your work shown) (c) What is the probability that someone likes bowling, given that they don't play a sport? Make sure to use proper notation. (Write any formulas used out entirely as part of your work shown) (d) Let A be the event that "someone prefers playing darts” and let B be the event that 'someone doesn't play a sport”. Are these events mutually exclusive? Explain why.

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The surface area of the box that Ian decorates is 562cm²

By looking at the shape, when you assemble this, it forms a cuboid which gives us:

The formula for surface area of cuboid is 2 (lb + bh + hl). By putting the values from 1 ,2,3, we get surface area is:

= 2*(15*8 + 8*7+ 7*15)

= 2*(120+56+105)

= 2*(120+161)

= 2*(281)

= 562cm².

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The inverse Laplace transform of the expressions are

f(t) = (3/2)sin(2t)

[tex]f(t) =\frac{4}{2!} t^2 e^t[/tex]

[tex]f(t) = e^{-t} (cos(2t) + sin(2t))[/tex]

[tex]f(t) = e^t sin(t)[/tex]

Using the formula for the inverse Laplace transform of 1/(s² + a²), we have:

F(s) = 3 / (s² + 4)

f(t) = (3/2)sin(2t)

Using the formula for the inverse Laplace transform of n!/(s-a)ⁿ⁺¹, we have:

F(s) = 4 / (s-1)³

[tex]f(t) = =\frac{4}{2!} t^2 e^t[/tex]

We can write the denominator of F(s) as (s+1)² + 4², and then use the formula for the inverse Laplace transform of 1/(s-a)² + b²:

F(s) = (2s+2) / (s² + 2s + 5)

[tex]f(t) = e^{-t} (cos(2t) + sin(2t))[/tex]

We can write the denominator of F(s) as (s-1)²+ 1, and then use the formula for the inverse Laplace transform of 1/(s-a)² + b²:

F(s) = (2s+1) / (s² - 2s + 2)

[tex]f(t) = e^t sin(t)[/tex]

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4x² + y = 17 can be evaluated when x = 2 and y = 1. dx² as dx² = (-16)² = 256.

To evaluate dx² when x=2 and y=1, we need to first differentiate the given equation with respect to x.

Differentiating 4x² + y = 17 with respect to x, we get:

8x + dy/dx = 0

Rearranging for dy/dx, we get:

dy/dx = -8x

Now we can substitute the given values of x and y to find dy/dx at x=2 and y=1:

dy/dx = -8(2) = -16

Finally, to find dx², we square the value of dy/dx:

dx² = (-16)² = 256

Therefore, dx² is equal to 256 when x=2 and y=1.

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The solution to each of the given simultaneous equations using graphical method is:

1) The solution is (1, 2)

2) The solution is (2/3, 5/3)

3)The solution is (0, 2)

4) The solution is: (3, 0)

5) There is no solution.

6) There is no solution.

There are three primary ways used to solve simultaneous equations and they are:

1) Elimination Method

2) Graphical Method

3) Substitution Method.

In this case, we are told to solve the simultaneous equations by graphical method and we have as attached:

1) y = x + 1

x + y = 3

The solution is (1, 2)

2) y = x + 1

x + 2y = 4

The solution is (2/3, 5/3)

3) y = x + 2

x + y = 2

The solution is (0, 2)

4) x + y = 3

x = 3

The solution is: (3, 0)

5) y = x + 4

y = x + 3

There is no solution as they are both parallel to each other

6) y = -x - 2

3y = -3x - 6

There is no solution as they are both parallel to each other

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False. It is not possible to determine the value of S4 for the sequence {(-1)" (n + 1)!} without knowing the first 4 terms of the sequence.

It seems like you have a question about the sequence {(-1)^n (n + 1)!} and whether S4 = 100. To answer this question, we need to evaluate the first four terms of the sequence and check if their sum (S4) equals 100.

1st term: (-1)^1 (1+1)! = -2!
2nd term: (-1)^2 (2+1)! = 3!
3rd term: (-1)^3 (3+1)! = -4!
4th term: (-1)^4 (4+1)! = 5!

Now, let's calculate the factorials and sum them up:

1st term: -2 = -2
2nd term: 6
3rd term: -24
4th term: 120

S4 = -2 + 6 - 24 + 120 = 100

So, the statement is true: for the sequence {(-1)^n (n + 1)!}, S4 = 100.

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Answer:

cannot be determined

Step-by-step explanation:

The sample mean used to construct the confidence interval (2.4, 9.8) cannot be determined without the confidence level. The confidence interval provides a range of possible values for the true population parameter (in this case, the population mean), based on the sample data and a chosen confidence level. The sample mean itself is not provided in the confidence interval, but rather the range within which the true population mean is likely to fall with a certain level of confidence. Therefore, without knowing the confidence level used to construct the interval, we cannot determine the sample mean.

The lower value of confidence level is approximately 314.67, under the condition that assuming x is normally distributed , then compute a 90% confidence interval to estimate μ.

To evaluate the sample mean

X' = (313 + 320 + 319 + 340 + 325 + 310 + 321 + 329 + 317 + 311 + 307 + 318) / 12

= 319.5

We have to evaluate the sample standard deviation

s = √([ (313 - 319.5)² + (320 - 319.5)²+ ... + (318 - 319.5)² ] / (12 - 1))

= 10.15

Now let us calculate the standard error

SE = s / √(n)

= 2.94

Then, the calculated the margin of error:

ME = Z x SE = 1.645 x 2.94

= 4.83

Let us calculate the confidence interval

There are two cases now

Cl = X' - ME = 319.5 - 4.85 = 314.67

Cl = X'+ ME = 319.5 + 4.85 = 324.35

Then, the lower value of confidence level is approximately 314.67.



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Answer:

Big O notation is used to describe the performance or complexity of an algorithm. Here are four rules to keep in mind when working with Big O notation:

Coefficients do not matter: When analyzing the performance of an algorithm, the coefficients of the terms in the Big O expression are not important. For example, O(2n) and O(n) are equivalent.

Ignore lower order terms: When analyzing the performance of an algorithm, only the highest order term is important. For example, O(n^2 + n) is equivalent to O(n^2).

Different inputs, different variables: When analyzing the performance of an algorithm with multiple inputs, use different variables to represent the size of each input. For example, if an algorithm takes two arrays as input, use n to represent the size of the first array and m to represent the size of the second array.

Worst case analysis: When analyzing the performance of an algorithm, consider the worst case scenario. For example, if an algorithm takes longer to run when the input is sorted in reverse order, use that scenario when calculating its Big O notation.

Step-by-step explanation:

Answer:

D. 33%.

Step-by-step explanation:

The gross margin (markup rate) can be calculated as follows:

Gross margin = (selling price - cost price) / selling price

In this case, the cost price of the skillet is $20 and the selling price is $30, so:

Gross margin = ($30 - $20) / $30

= $10 / $30

= 0.33 or 33%

Therefore, the answer is D. 33%.

Answer:

Dominic should pour 77 cubic inches of batter into the pan to leave enough space for the cornbread to rise in the oven.

Step-by-step explanation:

To find the volume of the batter that Dominic should pour into the pan, we need to calculate the volume of the pan and then subtract the volume of the space that needs to be left for the cornbread to rise.

The pan is in the shape of a rectangular prism, with dimensions of 11 inches (length) x 7 inches (width) x 2 inches (depth).

The total volume of the pan is:

V_total = length x width x depth

V_total = 11 in x 7 in x 2 in

V_total = 154 in³

To leave 1 inch of space at the top for the cornbread to rise, we need to subtract the volume of a rectangular prism with dimensions of 11 inches (length) x 7 inches (width) x 1 inch (height):

V_space = length x width x height

V_space = 11 in x 7 in x 1 in

V_space = 77 in³

Therefore, the volume of the batter that Dominic should pour into the pan is:

V = V_total - V_space

V = 154 in³ - 77 in³

V = 77 in³

So Dominic should pour 77 cubic inches of batter into the pan to leave enough space for the cornbread to rise in the oven.

Answer:

To find the volume of the batter, we need to first find the volume of the pan. We can use the formula for the volume of a rectangularsolid:

=lxwxh

where I is the length, w is the width, and h is the height.

In this case, the length of the pan is 11 inches, the width is 7 inches, and theheight is 1 inch (since the batter should only fill the pan up to 1 inch from the top).

So, V = 11 ? 7 ? 1 = 77 cubic inches.

Therefore, the volume of the batter that Dominic should pour into the pan is 77 cubic inches.

HOPE IT HELPS

HOPE IT HELPSPLEASE MARK ‼️ ME AS BRAINLIEST

The alternative hypothesis (H1) states that there is a significant difference, supporting the researcher's belief that the current ozone level is not at the normal level: H0μ = 4.4 ppm H1μ ≠ 4.4 ppm

The null hypothesis (H0) is that the current ozone level is at the normal level of 4.4 ppm. The alternative hypothesis (Ha) is that the current ozone level is not at the normal level of 4.4 ppm.

Step 2 of 6: Determine the level of significance (alpha).
Your answer: The level of significance (alpha) is 0.05.

Step 3 of 6: Identify the appropriate statistical test.
Your answer: Since we are comparing a sample mean to a population mean and the population standard deviation is unknown, we will use a t-test.

Step 4 of 6: Calculate the test statistic and p-value.
Your answer: The test statistic is calculated as (x - μ) / (s / √n), where x is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size. In this case, the test statistic is (4.8 - 4.4) / (0.5 / √10) = 2.83. Using a t-distribution table with 9 degrees of freedom (10 samples - 1), the two-tailed p-value is 0.018.

Step 5 of 6: Make a decision.
Your answer: Since the p-value of 0.018 is less than the level of significance of 0.05, we reject the null hypothesis. This means that there is evidence to support the alternative hypothesis that the current ozone level is not at the normal level of 4.4 ppm.

Step 6 of 6: Interpret the results.
Your answer: Based on the sample data, we can conclude with 95% confidence that the true population mean of ozone level is different from 4.4 ppm. The researcher's claim is supported by the data, and further investigation or action may be warranted to address the potential issue with the ozone level.
Step 1 of 6: State the null and alternative hypotheses.

The null hypothesis (H0) states that there is no significant difference between the observed data and the expected value, which in this case means that the current ozone level is at the normal level of 4.4 ppm. The alternative hypothesis (H1) states that there is a significant difference, supporting the researcher's belief that the current ozone level is not at the normal level.

H0: μ = 4.4 ppm
H1: μ ≠ 4.4 ppm

Here, μ represents the population mean ozone level. The null hypothesis assumes it is equal to 4.4 ppm, while the alternative hypothesis claims it is not equal to 4.4 ppm.

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The equivalent expression of the given expression is

The given expression is

1/3x + 3/4 + 2/3x _1/4 -2/3x

Combining like terms:

1/3x + 2/3x - 2/3x = 1/4 - 3/4

Simplifying further

(1/3x + 2/3x - 2/3x) = - (3/4 - 1/4)

Simplifying further:

1/3x = - 1/2

1/3x + 1/2.

Therefore, an expression equivalent to 1/3x + 3/4 + 2/3x - 1/4 - 2/3x is 1/3x + 1/2.

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(a) The maximum concentration occurs at t = 2.2 hours.

(b) The maximum concentration is 87 ng/mL.

(c) The maximum concentration is the highest point on the graph of the concentration versus time.

(d) The concentration is decreasing at the fastest rate at t = 1.3 hours.

(e) The concentration is decreasing at the fastest rate at the inflection point, which occurs at t = 1.3 hours and has a concentration of 65 ng/mL.

This is because the slope of the concentration versus time graph is steepest at the inflection point. (f) The concentration at the inflection point is 65 ng/mL. The inflection point is where the rate of change of the concentration transitions from decreasing to increasing.

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The evaluated integral is (x²/2) + C.

A nonsensical statement ("e7 - 248 Joke 1.- In (e? – 2x8) + C - 1 16 x").

It does not provide any meaningful information for evaluating the integral.

To evaluate the integral ∫x dx, we can use the power rule of integration, which states that:

∫x dx = (x²/2) + C,

C is the constant of integration.

Applying this rule, we get:

∫x dx = (x²/2) + C

The evaluated integral is (x²/2) + C.

The second part of the question seems to contain a joke or a nonsensical statement ("e7 - 248 Joke 1.- In (e? – 2x8) + C - 1 16 x").

It does not provide any meaningful information for evaluating the integral.

The integral in the problem is of the form ∫x dx, which can be evaluated using the power rule of integration, as mentioned in my previous answer. The second part of the question seems to contain a joke or a nonsensical statement that does not provide any meaningful information or context for the integral.

Without further information or clarification, it is not possible to provide a more detailed or specific solution.

If you have any additional information or context for the problem, please provide it so that I can assist you better.

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The diameter of the pizza for the largest slice, when the perimeter of the slice is 90 cm, should be 60 cm.

To maximize the area of your pizza slice, let the crust's length (arc) be 'a', and the two radii (straight edges) be 'r'. The given perimeter is 90 cm, so a + 2r = 90.

Since you want to maximize the slice's area, the angle between the two radii should be 180°, forming a semicircle. In a semicircle, the arc length is half the circumference of the circle, so a = (1/2) * πd (d=diameter).

Then, 2r = d, and thus a + d = 90. Replacing 'a' with (1/2) * πd gives (1/2) * πd + d = 90. Solving for 'd' results in d ≈ 60 cm, which is the diameter for the largest slice.

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We are not given the sample size, so we cannot calculate the test statistic without this information.

To find the value of the test statistic, we need to perform a hypothesis test using the given claim and poll results.

Null Hypothesis: p >= 0.5 (at least 50% of adults have a cell phone)

Alternative Hypothesis: p < 0.5 (fewer than 50% of adults have a cell phone)

where p represents the true proportion of adults who have a cell phone.

We are not given a significance level, so we will assume alpha = 0.05 for this test.

The test statistic for testing a proportion is calculated as:

z = (p_hat - p) / sqrt(p * (1 - p) / n)

where p_hat is the sample proportion, p is the hypothesized proportion, n is the sample size, and sqrt is the square root function.

We are given the sample proportion, but not the sample size or the hypothesized proportion. However, we can use the claim that fewer than 50% of adults have a cell phone to estimate the hypothesized proportion as 0.5 - d, where d is the deviation from 50% that represents "fewer than" 50%. Assuming a deviation of 5%, we can estimate the hypothesized proportion as 0.5 - 0.05 = 0.45.

Now we can substitute the given values into the formula for the test statistic:

z = (% who said they have a cell phone - 0.45) / sqrt(0.45 * (1 - 0.45) / n)

We are not given the sample size, so we cannot calculate the test statistic without this information.

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The 90% confidence interval for the population standard deviation is (0.249, 1.201). This means that we can be 90% confident that the population standard deviation falls between these two values.

To construct a 90% confidence interval for the population standard deviation, we can use the Chi-Square distribution.

First, we need to find the degrees of freedom, which is equal to the sample size minus 1. In this case, the degrees of freedom is 9.

Next, we can use the Chi-Square distribution table or calculator to find the critical values. For a 90% confidence interval with 9 degrees of freedom, the lower and upper critical values are 4.168 and 16.919, respectively.

Finally, we can plug in the sample standard deviation and the critical values into the formula for the confidence interval:

Lower bound = √((n-1)×s²/upper critical value) = √((10-1)×1²/16.919) = 0.249
Upper bound = √((n-1)×s²/lower critical value) = √((10-1)×1²/4.168) = 1.201

Therefore, the 90% confidence interval for the population standard deviation is (0.249, 1.201). This means that we can be 90% confident that the population standard deviation falls between these two values.

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The value of differentiate value of dx/dy -1/4.

Differentiate the equation y = x^2 + 5,

We can use the chain rule to find dx/dt at x = -2.

differentiate y with respect to t using chain rule

dy/dt = 2x dx/dt

substitute x = -2 and dy/dt = 5

5 = 2(-2) dx/dt

dx/dt = 5/(-4) = -5/4

To find dx/dy, we can start with the expression for dy/dx and solve for dx/dy using algebra.

y = x^2 + 5

dy/dx = 2x

dx/dy = 1/(dy/dx) = 1/(2x)

At x = -2,

dx/dy = 1/(2(-2)) = -1/4

Therefore, dx/dy = -1/4.

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The basic reproduction number Ro affects the dynamics of infectious diseases.

To solve for the relationship between i and s, we start with the given equations:

ds/dt = - Ro s(t) i(t)

di/dt = Ro s(t) i(t) – i(t)

We can rearrange the second equation to get:

di/dt + i = Ro s(t) i(t)

Then, we can use the chain rule to find di/ds:

di/ds = (di/dt) / (ds/dt)

di/ds = [Ro s(t) i(t) - i(t)] / [- Ro s(t) i(t)]

di/ds = (Ro s(t) i(t) - i(t)) / (-Ro s(t) i(t))

di/ds = -1/Ro + 1/(Ro s(t))

Now we have an expression for di/ds in terms of s and Ro. We can plot this relationship for different values of Ro to see how the dynamics of the disease change.

When Ro > 1, we have a situation where each infectious individual has the potential to infect more than one person, on average. In this case, we expect the fraction of infected individuals to increase as the fraction of susceptible individuals decreases. This means that the plot of di/ds will be negative, since the slope of the line will be decreasing as we move from left to right. This makes biological sense, since a larger Ro means that the disease is more easily transmitted and can spread quickly through a population.

On the other hand, when Ro < 1, we have a situation where each infectious individual infects fewer than one person, on average. In this case, we expect the fraction of infected individuals to decrease as the fraction of susceptible individuals decreases. This means that the plot of di/ds will be positive, since the slope of the line will be increasing as we move from left to right. This also makes biological sense, since a smaller Ro means that the disease is less easily transmitted and will have a harder time spreading through a population.

Overall, the model and the graphical depiction of di/ds make biological sense and help us understand how the basic reproduction number Ro affects the dynamics of infectious diseases.

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