Find the solution of the differential equation r'(t) = 5r(t) with the initial condition r(0) = (8, 2,6), where r(t) is a vector-valued function in three-space. (Use symbolic notation and fractions where needed. Give your answer in the form (x(t), y(t), z(t)).) r(t) =

Answer:

Because the cake pan holds 234 in.³ and the four round cake pans hold a total of around 226.19 in.³

Edit: I multiplied the diameter as the radius. Answer has now been corrected.

Step-by-step explanation:

To solve this, we need to figure out how much each batter can hold.

For this one, the formula is simple. we multiple everything together to get the volume.

13×9×2=234 in.³

For this one, you will have to use pi or π. A round cake pan is most likely a cylinder. Therefore, we will use the cylinder volume formula, which is:

V=πr²h

V=π(3)²(2)

V=π(9)(2)

V=18π

V is around 56.5486678 or 56.55. However, if you are using 3.14 for π, you will get something else, 56.52.

But remember that this is only one cake pan. There are 4 of them. So, we can simply multiple the number by 4. To help this be more accurate, I will go back to 18π first.

V=18π(4)

V=72π

=226.194671058

or around 226.19.

If you are using 3.14, just multiply by 3.14 instead of π. You will get a very similar result, with only a minor difference. It all depends on which one you are using.

Good luck with your homework! If this is correct, please give me brainliest :)

Answer:

Step-by-step explanation:

You want to know the volumes of a 13×9×2 inch rectangular cake pan, and of four 2-inch deep round cake pans 6 inches in diameter.

The volume of the rectangular cake pan is given by ...

  V = LWH

  V = (13 in)(9 in)(2 in) = 234 in³

The volume of four round cake pans with diameter d is given by ...

  V = 4×(π(d/2)²h) = πd²h

  V = π(6 in)²(2 in) = 72π in³ ≈ 226.2 in³

The rectangular pan holds more cake batter.

The cake pan holds 234 in³, and the four round pans hold 226.2 in³.

__

Additional comment

You often see the formula for the volume of a cylinder as ...

  V = πr²h

Since r = d/2, and the diameter is given here, we used the diameter in the formula above. We also made the formula apply to four (4) cake pans, which simplified our math.

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Answer: 5

Step-by-step explanation:

3x + 2 = 17

subtract 2 on both sides

3x + 2 - 2 = 17 - 2 > 3x + 15

Divide both sides by 3

[tex]\frac{3x}{3} = \frac{15}{3}[/tex]  = x=5

x = 5

The  new estimated demand is equal to the original estimated demand (ŷ) minus 12. This means that when the price is increased by 3 units, the estimated demand decreases by 12 units.

The equation ŷ = 70 - 4x represents a linear demand function for the product, where y is the estimated demand for the product and x is its price.

To answer the question, we can evaluate the change in demand when the price is increased by 3 units. We can do this by comparing the estimated demand at the original price (x) to the estimated demand at the new price (x + 3).

Original estimated demand:

ŷ = 70 - 4x

New estimated demand:

ŷ' = 70 - 4(x + 3) = 70 - 4x - 12 = ŷ - 12

Therefore, the new estimated demand is equal to the original estimated demand (ŷ) minus 12. This means that when the price is increased by 3 units, the estimated demand decreases by 12 units.

In other words, the demand for the product is negatively related to its price (as indicated by the negative coefficient of x in the demand function). When the price goes up, the estimated demand goes down, and vice versa. The magnitude of this effect is given by the coefficient of x, which in this case is 4. This means that for every one-unit increase in price, the estimated demand decreases by 4 units. Therefore, a 3-unit increase in price would lead to a decrease in estimated demand of 4 * 3 = 12 units.

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Answer: 1/nine.

Step-by-step explanation:To locate the probability of drawing a black marble after which a inexperienced marble, we need to keep in mind the range of approaches we can draw a black marble and then a inexperienced marble, and divide that with the aid of the whole range of methods we are able to draw  marbles with out replacement.

The opportunity of drawing a black marble on the primary draw is two/10, because there are 2 black marbles out of a total of 10 marbles. Then, in view that we draw the second one marble with out alternative, there will be nine marbles left within the bag. If we drew a black marble on the first draw, there could be 1 black marble and five inexperienced marbles left in the bag. So the possibility of drawing a green marble on the second draw, for the reason that a black marble became drawn on the first draw, is five/nine.

The chance of drawing a black marble after which a inexperienced marble is the manufactured from the chance of drawing a black marble on the primary draw and the opportunity of drawing a inexperienced marble on the second draw, given that a black marble turned into drawn on the primary draw. So we multiply 2/10 by five/nine to get 1/nine.

Therefore, the chance of drawing a black marble after which a inexperienced marble is 1/nine.

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Rounding to the nearest person, we estimate the city's population in 2025 to be 303,977 based on rate proportional.

When two quantities are directly proportional to one another with regard to time or another variable, this circumstance is referred to as being "rate proportional" in mathematics. For instance, if a population's rate of growth is proportionate to its size, the population will increase in size at an increasingly rapid rate. Similar to this, if an object's speed and applied force are proportionate, then increasing the force will increase an object's speed. Linear equations or differential equations can be used to describe proportional relationships, which are frequently found in many branches of science and mathematics.

To estimate the city's population in 2025, we can use the formula:

[tex]P(t) = P(0) * e^(kt)[/tex]

where P(0) is the initial population (123,580 in 1990), t is the time elapsed (in years), k is the growth rate (which we need to find), and P(t) is the population at time t.

To find k, we can use the fact that the population is increasing at a rate proportional to the existing population. This means that the growth rate (k) is constant over time. We can use the following formula to find k:

[tex]k = ln(P(t)/P(0)) / t[/tex]

where ln is the natural logarithm.

Plugging in the given values, we get:

k = ln(152,918/123,580) / 10 = 0.026

This means that the city's population is growing at a rate of 2.6% per year.

Now we can use the formula[tex]P(t) = P(0) * e^(kt)[/tex] to estimate the population in 2025:

[tex]P(35) = 123,580 * e^(0.026*35) = 303,977[/tex]

Rounding to the nearest person, we estimate the city's population in 2025 to be 303,977.

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(a) The set {(x1, x2)T | x1 + x2 = 0} forms a subspace of R2. This set contains the zero vector (0,0)T, is closed under vector addition, and is closed under scalar multiplication.

(b) The set {(x1, x2)T | x1x2 = 0} does not form a subspace of R2. Although it contains the zero vector (0,0)T and is closed under scalar multiplication, it is not closed under vector addition. For example, (1,0)T and (0,1)T are in the set, but their sum (1,1)T is not.

(c) The set {(x1, x2)T | |x1| = |x2|} does not form a subspace of R2. Although it contains the zero vector (0,0)T and is closed under vector addition, it is not closed under scalar multiplication.

For example, (1,1)T is in the set, but (2,2)T is not. Also, this set is not closed under vector addition since, for example, (1,0)T and (-1,0)T are in the set, but their sum (0,0)T is not.

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The radius of convergence is 7 and the interval does not include x = -7, the interval of convergence is [ -7, 7 ).

The radius of convergence of the series ∑[infinity]n=1(−1)nxn7√n is R = 7.

To find the radius of convergence, we can use the ratio test:

lim[n→∞] |(−1)^(n+1) * x^(n+1)/(7√(n+1))| / |(−1)^n * x^n/(7√n)|

= lim[n→∞] |x/(7√(n+1))|

= 0 for any finite x.

Therefore, the series converges for all x within a distance of 7 from 0. In other words, the radius of convergence is 7.

To find the interval of convergence, I, we need to check the endpoints x = -7 and x = 7 separately.

When x = -7, the series becomes ∑[infinity]n=1 (1/n)^(1/2), which is a harmonic series that diverges. Therefore, x = -7 is not in the interval of convergence.

When x = 7, the series becomes ∑[infinity]n=1 (-1)^n / n^(1/2), which converges by the alternating series test. Therefore, x = 7 is included in the interval of convergence.

Since the radius of convergence is 7 and the interval does not include x = -7, the interval of convergence is [ -7, 7 ).

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The probability that a person will be able to complete a 5000 mile trip without having to replace the car battery can be calculated using the cumulative distribution function (CDF) of the exponential distribution. The answer is approximately 60.7%.

To explain further, we know that the average value of the exponential distribution is 10000 miles. This means that the expected value of the number of miles a car can run before its battery wears out is 10000 miles. The probability that the car's battery will last for at least 5000 miles can be calculated using the CDF of the exponential distribution. The CDF of the exponential distribution is given by F(x) = 1 - e^(-x/μ), where x is the number of miles and μ is the average value of the distribution.

Substituting x = 5000 and μ = 10000, we get F(5000) = 1 - e^(-0.5) ≈ 0.607. Therefore, the probability that a person will be able to complete a 5000 mile trip without having to replace the car battery is approximately 60.7%. This means that there is a 60.7% chance that the car's battery will last for at least 5000 miles, and the person will not have to replace it during the trip.

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The 98.5% confidence interval for the proportion of cars with defects in the population is (0.065, 0.155). The interval does not include the claimed proportion of 0.08, indicating that the company's claim may not be accurate.

To calculate the confidence interval for this scenario, we can use the formula:

[tex]\begin{equation}CI = p \pm z \cdot \sqrt{\frac{p \cdot (1 - p)}{n}}\end{equation}[/tex]

where p is the sample proportion (the proportion of cars with defects in the sample), z is the z-score associated with the desired significance level, and n is the sample size.

In this case, the sample proportion is 33/300 = 0.11, which is higher than the claimed proportion of 0.08. We want to determine the confidence interval at a significance level of 0.015, which corresponds to a z-score of approximately 2.33.

Plugging in the values, we get:

[tex]\begin{equation}CI = 0.11 \pm 2.33 \cdot \sqrt{\frac{0.11 \cdot (1 - 0.11)}{300}}\end{equation}[/tex]

Simplifying the expression, we get:

CI = 0.11 ± 0.045

Therefore, the 98.5% confidence interval for the proportion of cars with defects in the population is:

CI = (0.065, 0.155)

This means that we are 98.5% confident that the true proportion of cars with defects in the population falls within this interval. Since the interval does not include the claimed proportion of 0.08, we have evidence to suggest that the company's claim may not be accurate.

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The value cosine of angle P is 4/√71

The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

Sin(tetha) = opp/hyp

cos(tetha) = adj/hyp

tan(tetha) = opp/adj

Here, the hyp is √71 and the opp is √55 , therefore using Pythagoras theorem

adj² = hyp²-opp²

= √71)²-√55)²

= 71-55

adj = √ 16

= 4

therefore the value cosine of P = adj/hyp

= 4/√71

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Answer: 61.7° (3 significant figures)

Step-by-step explanation:

The formula for cosine is cos(P) = [tex]\frac{A}{H}[/tex] . So, we need to find the length of side PR first before we can start, because it is the side that is adjacent (A) to angle P. To find the side PR we are going to use Pythagoras' theorem.

[tex]a^{2} =c^{2} -b^{2} \\\\a=\sqrt{c^{2} - b^{2} } \\\\a=\sqrt{(\sqrt{71} )^{2}-(\sqrt{55} )^{2} } \\\\a=\sqrt{71-55} \\\\a=\sqrt{16}\\\\a=4\\[/tex]

Now we can work out the cosine of angle P:

[tex]cos(P)=\frac{A}{H} \\\\cos(P)=\frac{4}{\sqrt{71} } \\\\P=cos^{-1} (\frac{4}{\sqrt{71} })\\\\P=61.6593...\\\\[/tex]°

[tex]P=61.7\\[/tex] °  (3 significant figures)

I hope this helps!

12[tex]x^{2}[/tex] + 18x - 12

Before we get started, notice that 12 and 18 are both divisible by 6. This will be helpful when factoring!

Part A:

factor out 6

6(2[tex]x^{2}[/tex] + 3x -2)

So I need to figure out 2 numbers that multiply together and make -2 (only -2 and 1 OR 1 and -2) and also notice that we have a coefficient of 2 on that x^2, so... honestly this is sort of a game of guess & check, but we will have 1 set of parentheses with + and another with - because the middle term is positive and the last term is negative.

Now rewrite:

6(2x-1)(x+2)

Part B:

Let's check by expanding:

6(2x-1)(x+2)

6(2[tex]x^{2}[/tex]+4x-x-2)

Combine like terms

6(2[tex]x^{2}[/tex]+3x-2)

Now multiply everything in parentheses times 6:

12[tex]x^{2}[/tex]+18x-12

which was the original equation, so our factored equation is correct.

The perimeter of the enlarged rectangle garden will be 25.92 meters.

To calculate the perimeter of the enlarged rectangle garden, we need to first find the new length and width after increasing each by 20%.

New length = 5.4 + (20% of 5.4) = 6.48 meters

New width = 1.5 + (20% of 1.5) = 1.8 meters

Now we can calculate the perimeter by adding up the lengths of all sides:

Perimeter = 2*(length + width)

Perimeter = 2*(6.48 + 1.8)

Perimeter = 2*(8.28)

Perimeter = 16.56 meters

Therefore, the perimeter of the enlarged garden will be 16.56 meters.

To solve this problem, we need to understand the basic formula for finding the perimeter of a rectangle, which is P = 2(l + w), where P is the perimeter, l is the length, and w is the width.

Given that the length of the garden is 5.4 meters and the width is 1.5 meters, we can find the initial perimeter by plugging these values into the formula:

P = 2(5.4 + 1.5)

P = 2(6.9)

P = 13.8 meters

Next, we are told that both the length and width will be increased by 20%. To find the new length and width, we can use the formula:

New length = length + (20% of length)

New width = width + (20% of width)

Plugging in the initial values, we get:

New length = 5.4 + (0.25.4) = 6.48 meters

New width = 1.5 + (0.21.5) = 1.8 meters

Now, we can use the same perimeter formula to find the new perimeter:

P = 2(new length + new width)

P = 2(6.48 + 1.8)

P = 2(8.28)

P = 16.56 meters

Therefore, the perimeter of the enlarged garden will be 16.56 meters.

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The value of the integral of the function g(x, y, z) = xyz over the surface of the rectangular solid is 0.

The integral of the function g(x, y, z) = xyz over the surface of the rectangular solid cut from the first octant by the planes x = a, y = b, and z = c.

To evaluate this integral, we first need to parameterize the surface of the rectangular solid. The surface of the rectangular solid can be parameterized as follows:

x = u

y = v

z = w

where u, v, and w are the parameters and 0 ≤ u ≤ a, 0 ≤ v ≤ b, and 0 ≤ w ≤ c.

Next, we need to find the partial derivatives of x, y, and z with respect to u and v:

∂x/∂u = 1

∂x/∂v = 0

∂y/∂u = 0

∂y/∂v = 1

∂z/∂u = 0

∂z/∂v = 0

Using the cross product of the partial derivatives, we can find the surface area element dS:

dS = (∂r/∂u) x (∂r/∂v) du dv = (i * j * k) du dv = k du dv

Now, we can integrate the function g(x, y, z) = xyz over the surface of the rectangular solid:

∫∫ g(x, y, z) dS = ∫∫ g(u, v, w) |k| du dv = ∫∫ uwbcos(π/2) du dv = 0

Therefore, the value of the integral of the function g(x, y, z) = xyz over the surface of the rectangular solid is 0.

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After expansion the expression is,

⇒ 65x/4 + 39/16

We have to given that;

Expression is,

⇒ 13/4 (5x + 3/4)

Now, We can simplify the expression by expansion,

⇒ 13/4 (5x + 3/4)

⇒ 5x × 13/4 + 13/4 × 3/4

⇒ 65x/4 + 39/16

Thus, After expansion the expression is,

⇒ 65x/4 + 39/16

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If the message P is not relatively prime to the enciphering modulus n=pq used in an RSA cipher, then P must share a common factor with either p or q, or both.

Let's assume that P shares a common factor with p (the case with q is symmetric).

If P shares a factor with p, then we can write:

P = a * p

where a is some integer. Since we know that P is not relatively prime to n=pq, it follows that p and q must share a common factor as well. Let's denote this common factor by d.

Then, we can write:

p = d * p'

q = d * q'

where p' and q' are relatively prime, and d is the greatest common divisor of p and q.

Substituting these expressions into n=pq, we get:

n = d^2 * p' * q'

Now, since P=a*p, we can rewrite this as:

P = a * d * p'

We know the values of P and n, and we just computed d and p', so we can solve for q':

q' = n / (d * p')

And since p' and q' are relatively prime, we have factored n=pq as:

n = d^2 * p' * q'

n = d * p * q'

n = d * p' * q'

where d, p', and q' have been computed from P and n.

Therefore, if the cryptanalyst discovers a message P that is not relatively prime to the enciphering modulus n pq used in an RSA cipher, and P shares a factor with either p or q, or both, then the cryptanalyst can factor n using the method outlined above.

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If there is a positive correlation between X and Y, then the regression equation, Y = bX + a, will have b > 0.

In a regression equation, the value of 'b' represents the slope of the line. When the correlation between X and Y is positive, it means that as X increases, Y also increases. In this case, the slope 'b' will be greater than 0, indicating that the line has a positive incline. The values of 'a', which is the Y-intercept, could be either positive or negative depending on the data, and it doesn't affect the positive correlation.

So, the correct answer is b > 0.

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Answer:

2/7 × 2/7 ×2/7 is equivalent

The probability that Jack and Jill will end up in the same class is 19/59.

The principles of probability and counting are fundamental concepts in probability theory and combinatorics. They are used to solve problems that involve uncertain events and counting arrangements of objects, respectively.

The principles of probability include:

Sample space: The set of all possible outcomes of a random experiment.

Event: A subset of the sample space.

Probability: A measure of the likelihood of an event, expressed as a number between 0 and 1.

Here we have

A group of 60 students is randomly split into 3 classes of equal size.

All partitions are equally likely. jack and jill are two students belonging to that group

Let's assume that Jack is assigned to a class, say the first class.

There are 20 students in that class, and the remaining 40 students are split evenly between the second and third classes.

Since all partitions are equally likely, each of the 59 remaining students has an equal chance of being assigned to any of the two remaining classes.

Now, we want to know the probability that Jill is assigned to the same class as Jack. There are 19 other students in the first class besides Jack, so there are 19 possible students in that class that Jill can be assigned to.

Out of the remaining 59 students, there are 40 in the other two classes, so Jill has 40 possible students she can be assigned to if she is not in Jack's class.

Hence,

The probability that Jill is assigned to the same class as Jack

= 19/59

Therefore,

The probability that Jack and Jill will end up in the same class is 19/59.

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Function A is a better fit for the data because the points are clustered closer to the x-axis. The correct answer is option D.

As per function A,

Here, the residual plot with dots randomly distributed about the x-axis suggests that the model fits the data well.

As per function B,

Here, the points display a pattern, such as a U-shaped pattern, so the model is not a good match for the data.

As we know that a residual plot with dots clustered closer to the x-axis implies that the projected values are closer to the observed values, indicating that the model fits the data better.

Therefore, Function A is a better fit for the data because the points are clustered closer to the x-axis.

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The random selection of a sample to represent the proportions of a population divided by sex and size of school attended is called stratified random sampling.

Stratified random sampling is a sampling method used when a population is divided into subgroups or strata based on certain characteristics, such as sex and size of school attended. In this method, a random sample is taken from each subgroup proportionate to its size in the population. This ensures that each subgroup is represented in the sample and reduces the chance of sampling bias.

For example, if the population of high school seniors is divided into two strata based on sex and two strata based on size of school attended, there would be four subgroups. A random sample would then be taken from each subgroup to create a representative sample of the entire population.

Stratified random sampling is commonly used in research studies and surveys to ensure that the sample accurately represents the population being studied. It allows for more precise estimates and statistical inferences to be made about the population as a whole.

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Answer:

3:5

Step-by-step explanation:

24:40

24 cars without sunroof

24+16=40

40 total cars

reduce the ratio of 24:40

12:20

6:10

3:5

the answer is 3:5

The calculated value of x from the intersecting secants is (b) 1.6

From the question, we have the following parameters that can be used in our computation:

intersecting secants

Using the theorem of intersecting secants, we have the following equation

a * b = c * d

In this case, we have

a = AE = 2

b = AB = 8

c = x

d = 10

Substitute the known values in the above equation, so, we have the following representation

2 * 8 = x * 10

Divide both sides by 10

x = 1.6

Hence, the value of x is 1.6

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Answer:

20,000

Step-by-step explanation:

Answer: 20,000

Step-by-step explanation:

The question wants you to round to the ten thousandths place, so you need to look in that place (which is the number 1 in this case).

The number to the right of 1 is equal to 5, so it should be rounded up to 2.

Answer:

5/2 x 4 or 2.5 x 4 because they both equal the same thing so yeah and the second one because if u convert the fraction to decimal that’s what it equals bye have a great day please give brainliest I’m new to this app

Step-by-step explanation:

The given transformations of reflection across the x-axis, followed by a right shift of 6 units, and a downward shift of 1 unit can be represented by the function:

f(x) = -(x - 6) - 1

A shift is a rigid translation in that it does not change the shape or size of the graph of the function.

The given transformations of reflection across the x-axis, followed by a right shift of 6 units, and a downward shift of 1 unit can be represented by the function:

f(x) = -(x - 6) - 1

This function reflects the original function across the x-axis, shifts it 6 units to the right, and then downward by 1 unit.

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The advantages of an open question include:

An open question is designed to elicit a broad and unrestricted response from the respondent. It encourages them to think creatively and explore various possibilities, which can lead to the introduction of new solutions. Unlike closed-ended questions that limit respondents to predefined options, an open question provides the freedom to express ideas, perspectives, and insights that may not have been considered before.

Furthermore, an open question allows the respondent to go in-depth with their answer. It prompts them to provide detailed explanations, examples, and personal experiences, allowing for a richer and more nuanced understanding of their thoughts and perspectives. This depth of response can unveil valuable insights, uncover underlying motivations, and provide context that may not have been captured with closed-ended questions. It also encourages active engagement and reflection from the respondent, as they are encouraged to express their thoughts and feelings in a more comprehensive manner.

Overall, open questions promote creativity, critical thinking, and a deeper exploration of ideas, making them advantageous in various contexts such as research, interviews, surveys, and problem-solving discussions.

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To find the length of the curve given by x = 7 + 9t^2, y = 6 + 6t^3, we can use the formula for arc length. The exact length of the curve is 30√5 - 6.

L = ∫(a to b) √[dx/dt]^2 + [dy/dt]^2 dt

where a and b are the endpoints of the parameter t.

Taking the derivatives of x and y with respect to t, we get:

dx/dt = 18t

dy/dt = 18t^2

Substituting into the formula for arc length, we get:

L = ∫(0 to 2) √[(18t)^2 + (18t^2)^2] dt

L = ∫(0 to 2) √(324t^2 + 324t^4) dt

L = ∫(0 to 2) 18t√(1 + t^2) dt

We can use u-substitution by setting u = 1 + t^2, du/dt = 2t, and solving for dt to get:

dt = du/(2t)

Substituting this into the integral, we get:

L = ∫(1 to 5) 9√u du

Using the power rule of integration, we get:

L = [6u^(3/2)]_1^5

L = 6(5√5 - 1√1)

L = 30√5 - 6

Therefore, the exact length of the curve is 30√5 - 6.

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Therefore, the composition of two rotations in a plane can lead to a translation, a rotation, or a reflection, depending on the relative orientations of the rotation axes.

The composition of two rotations in a plane can result in three types of non-identity planar isometries: a translation, a rotation, or a reflection.

Translation: If the axes of rotation are parallel, the composition of two rotations will result in a translation. In this case, the combined effect of the rotations is equivalent to a single translation in a specific direction.

Rotation: If the axes of rotation intersect at a point, the composition of two rotations will result in a single rotation about that point. The combined effect of the rotations will produce a new rotation with a different angle.

Reflection: If the axes of rotation are perpendicular, the composition of two rotations will result in a reflection. The combined effect of the rotations is equivalent to a single reflection across a line.

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Therefore, Kenji's profit is maximized at 5 shirts with a marginal cost of $12. The profit-maximizing quantity corresponds to the intersection of the marginal cost and marginal revenue curves.

Kenji's profit is maximized when he produces 5 shirts. At this level of production, the marginal cost of the last shirt he produces is $12, which is less than the price Kenji receives for each shirt he sells. The marginal cost of producing an additional shirt is $18, which is more than the price Kenji receives for each shirt he sells. Therefore, Kenji's profit-maximizing quantity corresponds to the intersection of the marginal cost and marginal revenue curves. Because Kenji is a price taker, this last condition can also be written as producing where marginal cost equals price.


Therefore, Kenji's profit is maximized at 5 shirts with a marginal cost of $12. The profit-maximizing quantity corresponds to the intersection of the marginal cost and marginal revenue curves.

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The probability of getting at least two students who reply "yes" is P(X ≥ 2) = 1 - P(X < 2) ≈ 1 - 0.0004 ≈ 0.9996

Rounding to four decimal places, the probability is 0.9996.

What is probability?

Probability is a branch of mathematics that deals with the study of random events or phenomena. It is the measure of the likelihood that an event will occur or not occur, expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

This is a binomial probability problem, since we are interested in the number of students out of a sample of 9 who reply "yes" to the question. Let X be the number of students who reply "yes".

Then X has a binomial distribution with n = 9 and p = 0.712, since each student's response is either "yes" or "no", and the probability of a "yes" response is 0.712.

We want to find the probability that at least two students out of the sample reply "yes". This can be written as:

P(X ≥ 2) = 1 - P(X < 2)

To calculate P(X < 2), we need to find the probabilities of X = 0 and X = 1, and add them together. We can use the binomial probability formula to find these probabilities:

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

where (n choose k) is the binomial coefficient, which gives the number of ways to choose k items from a set of n items.

Using this formula, we find:

P(X = 0) = (9 choose 0) * 0.712⁰ * (1-0.712)⁽⁹⁻⁰⁾ ≈ 0.000007

P(X = 1) = (9 choose 1) * 0.712¹ * (1-0.712)⁽⁹⁻¹⁾ ≈ 0.0004

Adding these probabilities together, we get:

P(X < 2) ≈ 0.0004 + 0.000007 ≈ 0.0004

Therefore, the probability of getting at least two students who reply "yes" is P(X ≥ 2) = 1 - P(X < 2) ≈ 1 - 0.0004 ≈ 0.9996

Rounding to four decimal places, the probability is 0.9996.

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