True or False: the number of true arithmetical statements involving positive integers, +, x,(,) and = is countable, i.e. "(17+31) x 2 = 96". Defend your answer.

Answer:

Step-by-step explanation:

is a square, all sides congruent, we add up and we have the perimeter

Perimeter = 4 + 4 + 4 + 4 = 16 m

The result of the perimeter is 16 meters (m).

To solve, we must first know that the perimeters in this problem should only be added to each side, which is 4, where it gives a result of 16 meters (m).

First of all we must know that in geometry, the perimeter is the sum of all the sides. A perimeter is a closed path that encompasses, surrounds, or skirts a two-dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.

With this we can say that the perimeters are those that are added from each side, so, what we need to do in this problem is just just add each side, each side is four, so we can add it by 4 since it asks us for that.

[tex] \bold{4 + 4 + 4 + 4 = \boxed{ \bold{16m}}}[/tex]

But we also have another step to solve this problem, which is just squaring it where it also gives us the same result, let's see:

[tex] \bold{2 {}^{4} = \boxed{ \bold{16 \: meters \: (m)}}}[/tex]

So, as we see, each resolution gives us the same result, therefore, the result of the perimeter is 16 meters (m).

In the topical method, we focus on discussing different topics or subjects by considering various aspects and details related to them. This approach allows us to think about and communicate more effectively by addressing both general and specific points that might be relevant to the conversation.

The method of communication that involves reviewing possible things (both general and specific) that might be said. The correct answer is: e. Topical.

The method described in the question is a form of "topical" communication. This approach involves considering different topics or subjects that may need to be discussed and organizing thoughts and information around them. By reviewing possible things that may be said on each topic, one can prepare for a more effective and focused communication.

                             This method can be especially helpful in situations where there are multiple topics to cover or when discussing complex information that requires careful organization and planning.

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The inverse of 'a' modulo 'm' is not possible for the given pairs (your day and month of birth: a=34, m=91) because 'a' and 'm' are not relatively prime.

To find the inverse of 'a' modulo 'm', we need to determine a number 'x' such that (a * x) % m = 1. This means that 'x' is the multiplicative inverse of 'a' modulo 'm'. However, for an inverse to exist, 'a' and 'm' must be relatively prime, meaning they do not have any common factors other than 1. In the given pair (a=34, m=91), 'a' and 'm' share a common factor of 13. Therefore, an inverse does not exist.

When 'a' and 'm' are not relatively prime, there is no integer 'x' that satisfies the equation (a * x) % m = 1. In this case, we cannot find the inverse of 'a' modulo 'm'. It is important to note that for an inverse to exist, 'm' must be a positive integer greater than 1, and 'a' must be a positive integer less than 'm'. In the given pair (34, 91), both conditions are met, but the lack of relative primality between 'a' and 'm' prevents the existence of an inverse.

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The nth term test, also known as the Test for Divergence, is a useful tool for determining the divergence of a given series. All of the given series - A) arctan(n), B) 61, C) n(n+3)/(n+4), and D) ln(n) - diverge according to the nth term test.

In order to use this test, you should analyze the limit of the sequence's terms as n approaches infinity. If the limit is not zero, then the series diverges.
For each of the series provided, let's apply the nth term test:
A) arctan(n), n=1 to infinity:
The limit as n approaches infinity of arctan(n) is π/2, which is not zero. Therefore, the series diverges.
B) 61:
Since the series consists of a constant term, the limit as n approaches infinity is 61, which is not zero. Therefore, the series diverges.
C) n(n+3)/(n+4), n=1 to infinity:
As n approaches infinity, the limit of n(n+3)/(n+4) is 1, which is not zero. Therefore, the series diverges.
D) ln(n), n=1 to infinity:
The limit as n approaches infinity of ln(n) is infinity, which is not zero. Therefore, the series diverges.
In conclusion, all of the given series - A) arctan(n), B) 61, C) n(n+3)/(n+4), and D) ln(n) - diverge according to the nth term test.

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Answer:y=-4/2x

Step-by-step explanation:

Answer:

Step-by-step explanation:

I think it would be 500

Answer:

The value of the car after 5 years is $12271.05

The present value of the car, PV = $16500

The rate of depreciation, r = 5.75%

r = 5.75/100

r = 0.0575

Step-by-step explanation:

Using the limit comparison test by comparing it to the series ∑(n=1 to ∞) 1/n^p, for convergence where p is a positive constant if p > 1, the series ∑(n=1 to ∞) n/√(n^3 + 4) converges. Otherwise, it diverges.

Let's determine the value of p to compare the given series:

Taking the limit as n approaches infinity of the ratio between the terms of the two series:

lim(n→∞) (n/√(n^3 + 4)) / (1/n^p)

Simplifying the expression inside the limit:

lim(n→∞) (n/n^p) / √(n^3 + 4)

Taking the reciprocal of the denominator:

lim(n→∞) (n/n^p) * (1/√(n^3 + 4))

Now, let's simplify further by dividing both the numerator and denominator by n:

lim(n→∞) 1/n^(p-1) * (1/√(n^2 + 4/n))

Since the term 4/n approaches 0 as n approaches infinity, we have:

lim(n→∞) 1/n^(p-1) * (1/√n^2)

Simplifying inside the limit:

lim(n→∞) 1/n^(p-1) * (1/n)

Combining the terms:

lim(n→∞) 1/n^p

For the series to converge, the limit above must be finite and positive.

Let's analyze the cases for p:

If p > 1:

In this case, the limit is 0, indicating that the series ∑(n=1 to ∞) 1/n^p converges. Therefore, the given series ∑(n=1 to ∞) n/√(n^3 + 4) also converges.

If p ≤ 1:

In this case, the limit approaches infinity, indicating that the series ∑(n=1 to ∞) 1/n^p diverges. Therefore, the given series ∑(n=1 to ∞) n/√(n^3 + 4) also diverges.

In conclusion, if p > 1, the series ∑(n=1 to ∞) n/√(n^3 + 4) converges. Otherwise, it diverges

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To solve the problem of determining the diameter of a circle using the rope that is already used to make a square of side length 5 inches, the first thing is to find out the length of the rope required to make the square.

If x represents the length of the rope required to make the square, then the perimeter of the square would be 4 * 5 = 20 inches since it has four sides of equal length. Hence, 20 inches = x inches. The formula for the circumference of a circle is C = 2πr, where C is the circumference, π is a mathematical constant with a value of approximately 3.14, and r is the radius of the circle.

Since the rope's length was used to make the square, it can also be used to make the circle by bending it into the shape of a circle. The formula for the circumference of a circle is 2πr, where r is the radius. Since the diameter of a circle is twice the radius, the formula for the diameter of a circle can be obtained by multiplying the radius by 2. If the length of the rope required to make the circle is y, then we can write: C = 2πr = y inches. Since the length of the rope used to make the square is equal to 20 inches and the circumference of the circle is equal to the length of the rope, we can write: y = 20Therefore, 2πr = 20 inches Dividing both sides of the equation by 2π, we get:r = 20 / 2π = 3.18 inches. To get the diameter of the circle, we multiply the radius by 2, therefore: diameter = 2r = 2 * 3.18 = 6.36 inches. The diameter of the circle is 6.36 inches.

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The answer is option 13. 4 square centimeters.

Let's first find the length of the sides of each triangle. Since the perimeter of each triangle is 10 centimeters, and each triangle has 3 sides of equal length, the length of each side of the triangles is given by;

Side length = Perimeter ÷ Number of sides

= 10 ÷ 3= 3.33 (rounded to 2 decimal places)

The base of each triangle measures 2 centimeters, and the length of the side is 3.33 centimeters.

We can use the Pythagorean theorem to find the height of the triangles. Using Pythagorean theorem,

a² + b² = c²where a = 1, b = h and c = 3.33

From the formula above, we can find that:

h² = c² - a²

= 3.33² - 1²

≈ 10.77h

≈ √10.77

≈ 3.28

The area of each triangle is given by the formula;

Area = 1/2 x base x height

= 1/2 x 2 x 3.28

= 3.28 square centimeters (rounded to 2 decimal places)

Since there are 4 triangles, the total area of the plastic triangles on the spinner is approximately:

Total area = 4 x 3.28

= 13.12 square centimeters (rounded to 2 decimal places)

Therefore, the answer is option 13. 4 square centimeters.

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Todd must purchase 300 copies of his resume to have enough resumes for 245 schools.

Todd plans to send resumes to 245 schools, so he needs at least 245 copies of his resume.

The local printer sells copies in sets of 100

so Todd must purchase at least the nearest multiple of 100 that is greater than or equal to 245.

Divide 245 by 100

245 ÷ 100 = 2.45

The nearest multiple of 100 that is greater than or equal to 2.45 is 3. Therefore, Todd needs to purchase 3 sets of 100 copies.

3 sets × 100 copies = 300 copies

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B. sigma^infinity_k = 0 x^k + 3, and the interval of convergence is (-1, 1).

To find the power series representation for g(x), we need to rewrite g(x) in terms of the given geometric series.

Notice that g(x) can be written as:
g(x) = x^3/1 - x = x^3 * (1/1-x)

We can now substitute the formula for the geometric series to get:

g(x) = x^3 * sigma^infinity_k = 0 x^k
= sigma^infinity_k = 0 (x^3 * x^k)
= sigma^infinity_k = 0 x^(k+3)

Therefore, the power series representation for g(x) is:
sigma^infinity_k = 0 x^(k+3)

The interval of convergence of this series is the same as that of the geometric series, which is |x| < 1.

In interval notation, this can be written as (-1, 1).

Therefore, the correct answer is B. sigma^infinity_k = 0 x^k + 3, and the interval of convergence is (-1, 1).

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Answer:

$108.80

Step-by-step explanation:

89.25x0.06 = $5.36 tax

89.25 + 5.36= 94.61

94.61 x 0.15 = 14.91 tip

94.61 + 14.91 = 108.80 total

A faster way: 89.25*1.06*1.15=108.80

Answer :   ∫∫[Y > 3/X] (1/3) * (y^2/9) dx/dy.

(a) To calculate P(X/Y > 1), we need to find the probability that the ratio of X to Y is greater than 1.

The joint probability density function of X and Y, since they are independent, is given by f(X,Y) = fX(x) * fY(y).

Given that X has a Uniform distribution on (0, 3), the probability density function of X, fX(x), is:

fX(x) = 1/(3-0) = 1/3 for 0 < x < 3, and 0 otherwise.

The probability density function of Y, fY(y), is given as:

fY(y) = y^2/9 for 0 < y < 3, and 0 otherwise.

Now, we can calculate P(X/Y > 1) as follows:

P(X/Y > 1) = ∫∫[X/Y > 1] f(X,Y) dxdy

          = ∫∫[X > Y] fX(x) * fY(y) dxdy

          = ∫∫[X > Y] (1/3) * (y^2/9) dxdy

          = ∫[0,3] ∫[0,x] (1/3) * (y^2/9) dydx

          = (1/3) ∫[0,3] [(1/9) * (y^3/3)] evaluated from 0 to x dx

          = (1/3) ∫[0,3] (x^3/27) dx

          = (1/3) * [(1/108) * (x^4)] evaluated from 0 to 3

          = (1/3) * [(1/108) * (3^4 - 0^4)]

          = (1/3) * [(1/108) * 81]

          = 1/4.

Therefore, P(X/Y > 1) = 1/4.

(b) To calculate P(X + Y > 2), we need to find the probability that the sum of X and Y is greater than 2.

We can calculate this as follows:

P(X + Y > 2) = ∫∫[X + Y > 2] f(X,Y) dxdy

            = ∫∫[X > 2 - Y] fX(x) * fY(y) dxdy

            = ∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy.

To solve this integral, we can break it into two parts based on the range of Y:

For 0 < y < 2:

∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy = ∫[0,2] ∫[2-y,3] (1/3) * (y^2/9) dxdy.

For 2 < y < 3:

∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy = ∫[2,3] ∫[0,3] (1/3) * (y^2/9) dxdy.

Calculating these integrals will give us the desired probability.

(c) To calculate P(X * Y > 3), we need to find the probability that the product of X and Y is greater than 3.

Similarly, we can set up the

integral:

P(X * Y > 3) = ∫∫[X * Y > 3] f(X,Y) dxdy

            = ∫∫[Y > 3/X] fX(x) * fY(y) dxdy

            = ∫∫[Y > 3/X] (1/3) * (y^2/9) dxdy.

We can then evaluate this integral over the appropriate ranges to find the desired probability.

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The Fisher information for a Pareto(α) distribution is I(α) = nα² / (α - 1)².

To determine the Fisher information for a sample from a Pareto(α) distribution, follow these steps:

1. Recall the Pareto(α) probability density function (PDF): f(x) = αxᵃ⁺¹), where x ≥ 1 and α > 0.
2. Compute the log-likelihood function, L(α) = ln(f(x1,...,xn)) = ∑ ln(α) - (α+1)ln(xi) for i = 1 to n.
3. Differentiate L(α) with respect to α: dL/dα = ∑ (1/α) - ln(xi).
4. Differentiate dL/dα again: d²L/dα² = -∑ (1/α²).
5. The Fisher information is the negative expectation of the second derivative: I(α) = -E(d²L/dα²).
6. Apply the Pareto(α) distribution's expectation: I(α) = nα² / (α - 1)².

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Based on the provided past trials, it is not possible to accurately predict the exact number of times a coin will land TAILS up if flipped 300 more times.

The given past trials consist of four numbers: 50, 44, 132, and 6600. It is unclear whether these numbers represent the number of times the coin landed TAILS up or the number of total flips. Assuming they represent the number of times the coin landed TAILS up, we can calculate the average number of TAILS per flip.

The average number of TAILS in the provided past trials is (50 + 44 + 132 + 6600) / 4 = 1682.

However, using this average to predict the future outcomes is not reliable. Each coin flip is an independent event, and the outcome of one flip does not affect the outcome of another. The probability of landing TAILS on each flip remains constant at 0.5, assuming the coin is fair.

Therefore, in the absence of additional information or a clear pattern in the past trials, we cannot make an accurate prediction of the number of times the coin will land TAILS up in the next 300 flips.

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Answer:

21.5 cm

Step-by-step explanation:

a² + b² = c²

20² + 8² = c²

400 + 64 = c²

464 = c²

c = √464

c = 21.5

Answer: 21.5 cm

The expected value of XA is 0.35, and the variance of XA is 0.2275.

To find the expected value of XA, we simply multiply the probability of A occurring (0.35) by 1 (the indicator variable when A occurs) and add the product of the probability of A not occurring (1 - 0.35 = 0.65) and 0 (the indicator variable when A does not occur). So, E(XA) = 0.35 * 1 + 0.65 * 0 = 0.35.

To find the variance of XA, we need to calculate the probability of each outcome (0 or 1) and its squared difference from the expected value. The variance formula for an indicator variable is Var(XA) = P(A)(1 - P(A)). Therefore, Var(XA) = 0.35 * (1 - 0.35) = 0.35 * 0.65 = 0.2275.

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The first digit must be 1. The remaining seven ones must be either 0 or 1.

Therefore, there can be formed [tex]2^7=128[/tex] different numbers.

The appropriate symbols to use for the answer are: µ - z * (s / √n) < δ < µ + z * (s / √n)

To construct a confidence interval estimate for the mean body temperature of all healthy humans, we can use the symbol "µ" to represent the population mean.

A 99% confidence interval estimate for the mean body temperature can be represented as:

µ - z * (s / √n) < µ < µ + z * (s / √n)

In this expression:

"z" represents the critical value from the standard normal distribution corresponding to the desired confidence level (in this case, 99%).

"s" represents the sample standard deviation.

"n" represents the sample size.

Therefore, the appropriate symbols to use for the answer are:

µ - z * (s / √n) < δ < µ + z * (s / √n)

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The simplified expression √39(√6 + 7) is equal to √(39 * 6) + 7√(39 .

To simplify the expression √39(√6 + 7), we can follow these steps:

Step 1: Distribute the square-root (√) to both terms inside the parentheses:

  √39 * √6 + √39 * 7

Step 2: Simplify the square roots separately:

  √(39 * 6) + √(39 * 7)

Step 3: Calculate the products under the square roots:

  √234 + √273

Step 4: Simplify the square roots further:

  √(9 * 26) + √(9 * 3 * 3 * 3)

Step 5: Split the square root of a product into the product of the square roots:

  √9 * √26 + √9 * √(3 * 3 * 3)

Step 6: Simplify the square roots of perfect squares:

  3√26 + 3√(3 * 3 * 3)

Step 7: Multiply the numbers outside the square roots:

  3√26 + 9√3

Note that the simplified form is obtained by simplifying the square roots as much as possible and combining like terms.

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Panchito should mix 30 ml of cranberry juice with 120 ml of orange juice to make 150 ml of a 22% cranberry-orange juice blend.

To determine the amounts of cranberry juice and orange juice that Panchito should mix, we can set up a system of equations based on the given information. Let's assume Panchito mixes x ml of cranberry juice and y ml of orange juice.

The total volume of the mixture is 150 ml: x + y = 150.

The percentage of cranberry juice in the mixture is 22%: (0.05x) / 150 = 0.22.

Simplifying the second equation, we get:

0.05x = 0.22 * 150

0.05x = 33

x = 33 / 0.05

x = 660 ml

Substituting this value back into the first equation, we can solve for y:

660 + y = 150

y = 150 - 660

y = -510 ml

Since the solution for y is negative, it is not feasible. This indicates that there is no way to create a 22% cranberry-orange juice blend using a 5% cranberry juice and a 90% orange juice.

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The expression for the elastic curve using the coordinate x1 for 0 ≤ x1 ≤ a is given by:[tex]y = (w * x1^2) / (2 * e * i) + C1 * x1 + C2.[/tex]

To determine the expression for the elastic curve using the coordinate x1 for 0 ≤ x1 ≤ a, we need to consider the equation for the deflection of a beam under bending. The elastic curve describes the shape of the beam due to applied loads.

The equation for the elastic curve of a beam can be expressed as:

[tex]y = (w * x1^2) / (2 * e * i) + C1 * x1 + C2,[/tex]

where:

y is the deflection at coordinate x1,

w is the distributed load acting on the beam,

e is the modulus of elasticity of the material,

i is the moment of inertia of the beam's cross-sectional shape,

C1 and C2 are constants determined by the boundary conditions.

In this case, since we are considering 0 ≤ x1 ≤ a, the boundary conditions will help us determine the constants C1 and C2. These conditions could be, for example, the deflection at the supports or the slope at the supports. Depending on the specific problem, the values of C1 and C2 would be determined accordingly.

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Therefore, (a) ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars, (b) ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour, (c) Approximately 559 cars in the garage at t = 10.

(a) To find the number of cars entering the parking garage over the interval 0 ≤ t ≤ 10, we need to integrate the rate of cars entering the garage with respect to time. ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars.
(b) To find ′′(5), we need to differentiate the rate of cars leaving the garage with respect to time twice. ′′(t) = -65cos(0.281) and ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour. This value represents the rate of change of the rate of cars leaving the garage at t = 5.
(c) To find the number of cars in the parking garage at time t = 10, we need to subtract the total number of cars leaving the garage from the total number of cars entering the garage from t = 0 to t = 10. This gives approximately 559 cars in the garage at t = 10.


Therefore, (a) ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars, (b) ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour, (c) Approximately 559 cars in the garage at t = 10.

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A) is correct answer. The inequality that represents the range of possible pounds purchased is 0.41 < (3.40/x) < 0.50.

The inequality that represents the range of possible pounds purchased is as follows:

0.41 < (3.40/x) < 0.50.

Let's discuss the given problem step-by-step.

Sarah pays $3.40 for x pounds of bananas.

The cost of one pound of bananas is greater than $0.41 and less than $0.50.

Therefore, the cost of x pounds of bananas can be written as:

3.40 < x(0.50) and 3.40 > x(0.41)

⇒ 0.41x < 3.40 < 0.50x

⇒ 0.41 < (3.40/x) < 0.50

Hence, the inequality that represents the range of possible pounds purchased is 0.41 < (3.40/x) < 0.50.

The answer is option A.

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Let us assume that m is not prime. This means that there exists a prime factor p of m such that p ≤ √m. Since a is relatively prime to m, it must also be relatively prime to p.

Now, let's consider the order of a modulo p. We know that ordpa divides p-1, since p is prime. However, since a and p are relatively prime, we also know that ordpa cannot be equal to p-1, since this would imply that a is a primitive root modulo p, which is impossible since p is a prime factor of m and therefore does not have any primitive roots modulo p.
So, ordpa must divide p-1, but it cannot be equal to p-1. Therefore, ordpa must be strictly less than m-1 (since m has p as a factor, which means that m-1 has p-1 as a factor). However, we know that ordma = m-1. This means that ordpa cannot be equal to ordma.
This is a contradiction, since we assumed that ordma = m-1 and that ordpa divides m-1. Therefore, our initial assumption that m is not prime must be false. Therefore, m must be prime.
In conclusion, if m is a positive integer and a is an integer relatively prime to m such that ordma = m-1, then m must be prime.

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Thus, the obtained function using the integration by parts:  -1/17 t^2 e^-17t - ∫ (-2/17t^2 e^-17t) dt.

To evaluate the integral ∫ t^2 e^-17t dt using integration by parts, we will use the formula:
∫ u dv = uv - ∫ v du

where u and dv are functions of t that we choose appropriately. Let's choose:

u = t^2   (so that du/dt = 2t)
dv = e^-17t dt   (so that v = (-1/17)e^-17t)

Using these choices, we can find du and v:
du = 2t dt
v = (-1/17)e^-17t

Now, we can apply the integration by parts formula:
∫ t^2 e^-17t dt = t^2 (-1/17)e^-17t - ∫ 2t (-1/17)e^-17t dt

Simplifying this expression, we get:
∫ t^2 e^-17t dt = (-1/17) t^2 e^-17t + (2/17) ∫ te^-17t dt

To evaluate the new integral ∫ te^-17t dt, we will use integration by parts again. This time, we will choose:
u = t   (so that du/dt = 1)
dv = e^-17t dt   (so that v = (-1/17)e^-17t)

Using these choices, we can find du and v:
du = dt
v = (-1/17)e^-17t

Now, we can apply the integration by parts formula again:
∫ te^-17t dt = t (-1/17)e^-17t - ∫ (-1/17)e^-17t dt

Simplifying this expression, we get:
∫ te^-17t dt = (-1/17) te^-17t + (1/289) e^-17t

Substituting this result back into our original expression, we get:
∫ t^2 e^-17t dt = (-1/17) t^2 e^-17t + (2/17) ((-1/17) te^-17t + (1/289) e^-17t))

Simplifying this expression, we get:
∫ t^2 e^-17t dt = (-1/17) t^2 e^-17t - (2/289) te^-17t - (2/4913) e^-17t

Therefore, the correct answer is (b): -1/17 t^2 e^-17t - ∫ (-2/17t^2 e^-17t) dt.

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For any integer input x in the domain of relation a, if x is related to 2, then the output will be u2.

Based on the given information, we know that the domain of the relation a is the set of integers. Additionally, we know that 2 is related to y under relation a, with the output being u2.

Therefore, we can conclude that for any integer input x in the domain of relation a, if x is related to 2, then the output will be u2. However, we do not have enough information to determine the outputs for other inputs in the domain.

In other words, we know that the relation a contains at least one ordered pair (2, u2), but we do not know if there are any other ordered pairs in the relation.

The correct question should be :

In the given relation a, if an integer input x is related to 2, what is the corresponding output?

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Answer:

C

Step-by-step explanation:

Input x 6 = output for each of these numbers

3x6 =18

6x6 =36

11x6 = 66

12x6 = 72

the other options are incorrect. A is divided by 4, B is times 4, and D is divided by 6.

To find the area enclosed by the polar curve r = 12sinθ, we can use the formula for the area of a polar curve: A = 1/2 * ∫(r^2)dθ. For r = 12sinθ, the integral limits are from 0 to π because the curve covers a full period of the sine function.

Let's evaluate the integral using angle identity:

A = 1/2 * ∫(r^2)dθ
A = 1/2 * ∫((12sinθ)^2)dθ, with θ from 0 to π

A = 1/2 * ∫(144sin^2θ)dθ

Now, we can use the double angle identity sin^2θ = (1 - cos(2θ))/2:

A = 1/2 * ∫(144(1 - cos(2θ))/2)dθ

A = 72 * ∫(1 - cos(2θ))dθ, with θ from 0 to π

Now, we can integrate:

A = 72 * [θ - 1/2 * sin(2θ)] from 0 to π

A = 72 * [π - 0 - (1/2 * sin(2π) - 1/2 * sin(0))]

A = 72 * π

The exact area enclosed by the polar curve r = 12sinθ is 72π square units.

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this function assumes that the baseball is thrown from ground level, and it does not take into account any external factors that may affect the trajectory of the ball (such as air resistance, wind, or spin).

Assuming that air resistance can be ignored, the height (in feet) of a baseball thrown upward with an initial velocity of 30 ft/s at time t (in seconds) can be modeled by the function:

h(t) = 30t - 16t^2

This function represents the position of the baseball above the ground, and it is a quadratic equation with a downward-facing parabolic shape. The initial velocity of 30 ft/s corresponds to the coefficient of the linear term, and the coefficient of the quadratic term (-16) is half the acceleration due to gravity (32 ft/s^2).

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