Trapezium: Parallel side 1 is 8m Parallel side 2 is 10m and area is 126m square. What is the Height? Show your working.

The derivative of the function f(x) = (9[tex]x^{6}[/tex] + 8x³)³ is f'(x) = 4(9[tex]x^{6}[/tex] + 8x³)³(54x³ + 24x²).

To find the derivative of the function f(x) = (9x² + 8x³)³, you need to apply the Chain Rule. The Chain Rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, let u = 9x + 8x.

First, find the derivative of the outer function with respect to u: d( u³ )/du = 4u³.
Next, find the derivative of the inner function with respect to x: d(9x² + 8x³)/dx = 54x³ + 24x².

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The partial derivatives of u with respect to x, y, and t are, [tex]\dfrac{\partial u}{\partial x}[/tex] = 22, [tex]\dfrac{\partial u}{\partial y}[/tex] = 18 and [tex]\dfrac{\partial u}{\partial t}[/tex] = 40.

We can use the chain rule to find the partial derivatives of u with respect to x, y, and t.

First, we will find the partial derivative of u with respect to r and s:

u = r² + s²

[tex]\dfrac{\partial u}{\partial r}[/tex] = 2r

[tex]\dfrac{\partial u}{\partial s}[/tex] = 2s

Next, we will find the partial derivatives of r with respect to x, y, and t:

r = y + xcos(t)

[tex]\dfrac{\partial r}{\partial x}[/tex] = cos(t)

[tex]\dfrac{\partial r}{\partial y}[/tex] = 1

[tex]\dfrac{\partial r}{\partial t}[/tex] = -xsin(t)

Similarly, we will find the partial derivatives of s with respect to x, y, and t:

s = x + ysin(t)

[tex]\dfrac{\partial s}{\partial x}[/tex] = 1

[tex]\dfrac{\partial s}{\partial y}[/tex] = sin(t)

[tex]\dfrac{\partial s}{\partial t}[/tex] = ycos(t)

Now, we can use the chain rule to find the partial derivatives of u with respect to x, y, and t:

[tex]\dfrac{\partial u}{\partial x} = \dfrac{\partial u}{\partial r} \times \dfrac{\partial r}{\partial x} + \dfrac{\partial u}{\partial s} \times \dfrac{\partial s}{\partial x}[/tex]

[tex]\dfrac{\partial u}{\partial x}[/tex] = 2r * cos(t) + 2s * 1

At x = 4, y = 5, t = 0, we have:

r = 5 + 4cos(0) = 9

s = 4 + 5sin(0) = 4

Substituting these values into the partial derivative formula, we get:

[tex]\dfrac{\partial u}{\partial x}[/tex] = 2(9)(1) + 2(4)(1) = 22

Similarly, we can find the partial derivatives with respect to y and t:

[tex]\dfrac{\partial u}{\partial y} = \dfrac{\partial u}{\partial r} \times \dfrac{\partial r}{\partial y} + \dfrac{\partial u}{\partial s} \times \dfrac{\partial s}{\partial y}[/tex]

[tex]\dfrac{\partial u}{\partial y}[/tex] = 2r * 1 + 2s * sin(t)

[tex]\dfrac{\partial u}{\partial t}[/tex] = 2(9)(1) + 2(4)(0) = 18

[tex]\dfrac{\partial u}{\partial t} = \dfrac{\partial u}{\partial r} \times \dfrac{\partial r}{\partial t} + \dfrac{\partial u}{\partial s} \times \dfrac{\partial s}{\partial t}[/tex]

[tex]\dfrac{\partial u}{\partial t}[/tex] = 2r * (-xsin(t)) + 2s * (ycos(t))

[tex]\dfrac{\partial u}{\partial t}[/tex] = 2(9)(-4sin(0)) + 2(4)(5cos(0)) = 40

Therefore, the partial derivatives of u with respect to x, y, and t are:

[tex]\dfrac{\partial u}{\partial x}[/tex] = 22

[tex]\dfrac{\partial u}{\partial y}[/tex] = 18

[tex]\dfrac{\partial u}{\partial t}[/tex] = 40

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The average slope of f(x) on the interval [-6,4] is equal to f'(3.5) = -12(3.5) = -42.

The Mean Value Theorem (MVT) for a function f(x) on the interval [a,b] states that there exists a point c in (a,b) such that f'(c) = (f(b) - f(a))/(b - a).

In this problem, we are asked to find the average slope of the function f(x) = 3 - 6x² on the interval [-6,4]. The average slope is:

(f(4) - f(-6))/(4 - (-6)) = (3 - 6(4)² - (3 - 6(-6)²))/(4 + 6) = -42

So, we need to find a point c in (-6,4) such that f'(c) = -42. The derivative of f(x) is:

f'(x) = -12x

Setting f'(c) = -42, we get:

-12c = -42

c = 3.5

Therefore, the point c = 3.5 satisfies the conditions of the Mean Value Theorem, and the average slope of f(x) on the interval [-6,4] is equal to f'(3.5) = -12(3.5) = -42.

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Answer:22

Step-by-step explanation:

First you put

40/100

and that makes

11/22

The inverse Laplace transform of f(s) is: f(t) = -2t*u(t)[tex]e^{-t}[/tex] - 4u(t)[tex]e^{-t}[/tex]+ 4u(t)

What is convolution theorem?

The convolution theorem is a fundamental result in mathematics and signal processing that relates the convolution operation in the time domain to multiplication in the frequency domain.

To find the inverse Laplace transform of the given function, we will use the convolution theorem, which states that the inverse Laplace transform of the product of two functions is the convolution of their inverse Laplace transforms.

We can rewrite the given function as:

f(s) = 1/(s+1)² * (s² + 4)

Taking the inverse Laplace transform of both sides, we get:

[tex]L^{-1}[/tex]{f(s)} = [tex]L^{-1}[/tex]{1/(s+1)²} *[tex]L^{-1}[/tex]{s² + 4}

We can use partial fraction decomposition to find the inverse Laplace transform of 1/(s+1)²:[tex]e^{-t}[/tex]

1/(s+1)² = d/ds(-1/(s+1))

Thus, [tex]L^{-1}[/tex]{1/(s+1)²} = -t*[tex]e^{-t}[/tex]

To find the inverse Laplace transform of s²+4, we can use the table of Laplace transforms and the property of linearity of the Laplace transform:

L{[tex]t^{n}[/tex]} = n!/[tex]s^{(n+1)}[/tex]

L{4} = 4/[tex]s^{0}[/tex]

[tex]L^{-1}[/tex]{s² + 4} = L^-1{s²} + [tex]L^{-1}[/tex]{4} = 2*d²/dt²δ(t) + 4δ(t)

Now, we can use the convolution theorem to find the inverse Laplace transform of f(s):

[tex]L^{-1}[/tex]{f(s)} = [tex]L^{-1}[/tex]{1/(s+1)²} * [tex]L^{-1}[/tex]{s² + 4} = (-te^(-t)) * (2d²/dt²δ(t) + 4δ(t))

Simplifying this expression, we get:

[tex]L^{-1}[/tex]{f(s)} = -2[tex]te^{-t}[/tex]δ''(t) - 4[tex]te^{-t}[/tex]δ'(t) + 4[tex]e^{-t}[/tex]δ(t)

Therefore, the inverse Laplace transform of f(s) is:

f(t) = -2t*u(t)[tex]e^{-t}[/tex] - 4u(t)[tex]e^{-t}[/tex]+ 4u(t).

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Answer:

10,11

Step-by-step explanation:

Givne set: {7, 8 , 9 , 10 , 11}

To solve the inequality, isolate 'h'.

        2h - 3 > 15

Add 3 to both sides,

     2h - 3 + 3 > 15 + 3

               2h  > 18

Divide both sides by 2,

                [tex]\sf \dfrac{2h}{2} > \dfrac{18}{2}[/tex]

                 h > 9

h = {10 , 11}

The decay rate of f is approximately 4.0822% per unit of x.

To change [tex]f(x) = 40(0.96)^x[/tex] to an exponential function with base e, we can use the fact that:

[tex]e^{ln(a)} = a[/tex], where a is a positive real number.

First, we can rewrite 0.96 as[tex]e^{ln(0.96)}[/tex]:

[tex]f(x) = 40(e^{ln(0.96)})^x[/tex]

Then, we can use the property of exponents to simplify this expression:

[tex]f(x) = 40e^{(x*ln(0.96))}[/tex]

This is an exponential function with base e.

To approximate the decay rate of f, we can look at the exponent x*ln(0.96).

The coefficient of x represents the rate of decay. In this case, the coefficient is ln(0.96).

Using a calculator, we can approximate ln(0.96) as -0.040822. This means that the decay rate of f is approximately 4.0822% per unit of x.

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Answer:

b=7

Step-by-step explanation:

We know that for a parallelogram, The formula is a=bh

so plug it in

28=b4

Divide both sides by 4:

b=7

Answer:

b = 7 m

Step-by-step explanation:

the area (A) of a parallelogram is calculated as

A = bh ( b is the base and h the perpendicular height )

here h = 4 and A = 28 , then

28 = 4b ( divide both sides by 4 )

7 = b

There are [tex](1/2) * (2^{100} - 2)[/tex] partitions of {1, 2, ..., 100} into two parts.

We can use the following formula:

Number of partitions = (n choose k)/2, where n is the total number of elements, and k is the number of elements in one of the two parts.

In this case, we want to divide the set {1, 2, ..., 100} into two parts, each with k elements.

Since we are not distinguishing between the two parts, we divide the total number of partitions by 2.

The number of ways to choose k elements from a set of n elements is given by the binomial coefficient (n choose k).

So the number of partitions of {1, 2, ..., 100} into two parts is:

(100 choose k)/2

where k is any integer between 1 and 99 (inclusive).

To find the total number of partitions, we need to sum this expression for all values of k between 1 and 99:

Number of partitions = (100 choose 1)/2 + (100 choose 2)/2 + ... + (100 choose 99)/2

This is equivalent to:

Number of partitions = (1/2) * ([tex]2^{100}[/tex] - 2)

Therefore, there are (1/2) * ([tex]2^{100][/tex] - 2) partitions of {1, 2, ..., 100} into two parts.

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The time is 4.6 seconds when Sam enters the water again

So, we have the equation:

0 = -4.9t² + 4t + 10

Now, we can solve this quadratic equation for t using the quadratic formula:

t = (-b ± √(b² - 4ac)) / 2a

In our equation, a = -4.9, b = 4, and c = 10.

t = (-4 ± √(4² - 4(-4.9)(10))) / 2(-4.9)

t = (-4 ± √(16 + 196)) / (-9.8)

t = (-4 ± √212) / (-9.8)

The two possible values for t are:

t ≈ 0.444 (when Sam is at the surface of the water, just after jumping)

t ≈ 4.597 (when Sam enters the water again)

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Answer: The time is 4.6 seconds when Sam enters the water again

So, we have the equation:

0 = -4.9t² + 4t + 10

Now, we can solve this quadratic equation for t using the quadratic  formula:

t = (-b ± √(b² - 4ac)) / 2a

In our equation, a = -4.9, b = 4, and c = 10.

The two possible values for t are:

t ≈ 0.444 (when Sam is at the surface of the water, just after jumping)

t ≈ 4.597 (when Sam enters the water again)

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The farmer would need to sample at least 108 1-acre plots to estimate the mean yield per acre with a margin of error of 1 bushel and a 99% confidence level.

What is Standard deviation ?

Standard deviation is a measure of how spread out a set of data is from the mean (average) value. It tells you how much the individual data points deviate from the mean. A smaller standard deviation indicates that the data points are clustered closer to the mean, while a larger standard deviation indicates that the data points are more spread out.

a. To find the 99% confidence interval (CI) for the mean yield per acre, we can use the formula:

CI = X' ± Zα÷2 * σ÷√n

where X' is the sample mean, σ is the population standard deviation, n is the sample size, and Zα÷2 is the critical value for a 99% confidence level, which can be found using a standard normal distribution table or calculator.

Zα÷2 = 2.576 (from a standard normal distribution table for a 99% confidence level)

Substituting the given values, we get:

CI = 61.49 ± 2.576 * 3.754÷√25

CI = 61.49 ± 1.529

CI = (59.96, 63.02)

Therefore, we are 99% confident that the true mean yield per acre for the farmer using the organic method is between 59.96 and 63.02 bushels.

b. To find the sample size required to have a margin of error of 1 bushel and a 99% confidence level, we can use the formula:

n = (Zα÷2 * σ÷E)²

where Zα÷2 is the critical value for a 99% confidence level (2.576), σ is the population standard deviation (which we assume to be 3.754), and E is the desired margin of error (1 bushel).

Substituting the given values, we get:

n = (2.576 * 3.754÷1)²

n ≈ 108

Therefore, the farmer would need to sample at least 108 1-acre plots to estimate the mean yield per acre with a margin of error of 1 bushel and a 99% confidence level.

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For the cumulative distribution function, p(1 < X ≤ 2) ≈ 0.2222.

The probability p(1 < X ≤ 2) can be computed by finding the area under the curve of the probability density function (pdf) between x = 1 and x = 2.

Since the cumulative distribution function (cdf) is given, we can differentiate it to obtain the pdf. Thus, the pdf is:

f(x) = { 0, x < 0

18x, 0 ≤ x < 1/4

31/6 - 79x/12, 1/4 ≤ x < 2/3

0, x ≥ 2/3

The probability that 1 < X ≤ 2 can then be computed as follows:

p(1 < X ≤ 2) = ∫₁² f(x) dx

Using the pdf defined above, we can evaluate the integral as follows:

p(1 < X ≤ 2) = ∫₁^(2/3) (31/6 - 79x/12) dx

= [(31/6)x - (79/24)x^2]₁^(2/3)

= (31/6)(2/3) - (79/24)(4/9) - (0) (substituting x = 2/3 and x = 1)

= 0.2222

Therefore, p(1 < X ≤ 2) ≈ 0.2222.

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For the cumulative distribution function, p(1 < X ≤ 2) ≈ 0.2222.

The probability p(1 < X ≤ 2) can be computed by finding the area under the curve of the probability density function (pdf) between x = 1 and x = 2.

Since the cumulative distribution function (cdf) is given, we can differentiate it to obtain the pdf. Thus, the pdf is:

f(x) = { 0, x < 0

18x, 0 ≤ x < 1/4

31/6 - 79x/12, 1/4 ≤ x < 2/3

0, x ≥ 2/3

The probability that 1 < X ≤ 2 can then be computed as follows:

p(1 < X ≤ 2) = ∫₁² f(x) dx

Using the pdf defined above, we can evaluate the integral as follows:

p(1 < X ≤ 2) = ∫₁^(2/3) (31/6 - 79x/12) dx

= [(31/6)x - (79/24)x^2]₁^(2/3)

= (31/6)(2/3) - (79/24)(4/9) - (0) (substituting x = 2/3 and x = 1)

= 0.2222

Therefore, p(1 < X ≤ 2) ≈ 0.2222.

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The missing frequency for the interval 10-12 is 21.

Let's call the missing frequencies as x and y for the intervals 10-12 and 16-18 respectively.

We know that the total number of observations is 50 and the mean is 16.

To find x and y, we can use the formula for the mean of grouped data:

Mean = (sum of (midpoint of each interval * frequency)) / (total number of observations)

16 = ((11+13)5 + (17+19)3 + 1410 + 202 + 21*y) / 50

Simplifying the above equation, we get:

800 + 21y = 800

y = 0

This means that the missing frequency for the interval 16-18 is 0.

To find the missing frequency for the interval 10-12, we can use the fact that the total number of observations is 50:

x + 5 + 10 + 9 + 3 + 2 + 0 = 50

x = 21

Therefore, the missing frequency for the interval 10-12 is 21.

So the complete frequency table is:

Class Intervals Frequencies

10-12 5 + 21 = 26

12-14 ?

14-16 10

16-18 0

18-20 9

20-22 3

22-24 2

TOTAL 50.

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Among the following pairs of sets, I'll help you identify the ones that are equal:

1. {1, 3, 3, 3, 5, 5, 5, 5, 5} and {5, 3, 1}:

These sets are equal because in set notation, repetitions are not counted.

Both sets have the unique elements {1, 3, 5}.

2. {{1}} and {1, [1]}:

These sets are not equal because the first set contains a single element which is the set {1}, while the second set contains two distinct elements, 1 and [1]

(assuming [1] is a different notation for an element).

3. {0} and [1, 2]:

These sets are not equal because they have different elements. The first set contains the single element 0, while the second set contains the elements 1 and 2.

4. [[1], [2]]:

This is not a pair of sets, so it cannot be compared for equality.

In summary, the equal pair of sets among the given options is {1, 3, 3, 3, 5, 5, 5, 5, 5} and {5, 3, 1}.

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Using the simple interest system, the interest rate for which Gary deposited $9,000 and earned $180 in interest after four months is 6%.

The simple interest system is based on the process of computing interest on the principal only for each period.

This contrasts with the compound interest system that charges interest on both accumulated interest and the principal.

The simple interest formula is given as SI = (P × R × T)/100, where SI = simple interest, P = Principal, R = Rate of Interest in % per annum, and T = Time.

The principal amount invested by Gary = $9,000

Time = 4 months = 4/12 years

Interest = $180

Therefore, 180 = ($9,000 x R x 4/12)/100

R = 180/($9,000 x 4/12)/100

R = 6%

Thus, the interest rate is 6%.

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Answer:

7/2x+30

Step-by-step explanation:

(9/4x+6)-(-5/4x-24)

9/4x+6-(-5/4x)-(-24)

9/4x+6+5/4x+24

14/4x+30

7/2x+30

The probability density function is f(x) = (1/5)e^(-x/5) for x >= 0, the probability it will take the mechanic less than 6 minutes to change oil is 0.699

a. The probability density function (PDF) for the time it takes a mechanic to change the oil in a car, given that it follows an exponential distribution with a mean of 5 minutes, is:

f(x) = (1/5)e^(-x/5) for x >= 0

b. The probability that it will take a mechanic less than 6 minutes to change the oil is given by:

P(X < 6) = ∫0^6 f(x) dx

= ∫0^6 (1/5)e^(-x/5) dx

= [-e^(-x/5)]_0^6

= 1 - e^(-6/5)

≈ 0.699

c. The probability that it will take a mechanic between 3 and 5 minutes to change the oil is given by:

P(3 < X < 5) = ∫3^5 f(x) dx

= ∫3^5 (1/5)e^(-x/5) dx

= [-e^(-x/5)]_3^5

= e^(-3/5) - e^(-1)

≈ 0.181

d. The variance of the time it takes to change the oil can be calculated using the formula:

Var(X) = σ^2 = 1/λ^2

where λ is the rate parameter of the exponential distribution, which is the reciprocal of the mean. Therefore, in this case:

λ = 1/5

σ^2 = (1/λ)^2 = 5^2 = 25

So, the variance of the time it takes to change the oil is 25.

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The particular solution to the differential equation by the method of undetermined coefficients is [tex]y \_p(x) = (-6x^2 - 16x - 80) + e^{(2x)}(x^2 + x - 44).[/tex]

To solve this differential equation using the method of undetermined coefficients, we assume that the particular solution takes the form:

[tex]y \_ p(x) = (Ax^2 + Bx + C) + e^{(2x)}(Dx^2 + Ex + F)[/tex]

where A, B, C, D, E, and F are constants to be determined.

To determine the values of these constants, we differentiate y_p(x) four times and substitute the result into the differential equation. We get:

[tex]y \_p(x) = Ax^2 + Bx + C + e^{(2x)}(Dx^2 + Ex + F)[/tex]

[tex]y\_p'(x) = 2Ax + B + 2e^{(2x)}(Dx^2 + Ex + F) + 2e^{(2x)}(2Dx + E)[/tex]

[tex]y \_p''(x) = 2A + 4e^{(2x)}(Dx^2 + Ex + F) + 8e^{(2x)}(Dx + E) + 4e^{(2x)(2D)}[/tex]

[tex]y\_p''(x) = 8e^{(2x)}(Dx^2 + Ex + F) + 24e^{(2x)(Dx + E)} + 16e^{(2x)(D)}[/tex]

[tex]y \_p^4(x) = 32e^{(2x)(Dx + E) }+ 32e^{(2x)(D)}[/tex]

Substituting these into the original differential equation, we get:

[tex](32e^{(2x)(Dx + E)} + 32e^{(2x)(D))} - 2(8e^{(2x)}(Dx^2 + Ex + F) + 24e^{(2x)(Dx + E)} + 16e^{(2x)(D))} + (Ax^{2 }+ Bx + C + e^{(2x)}(Dx^2 + Ex + F))(x - 4)^2 = (x - 4)^2[/tex]

Simplifying this expression, we get:

[tex](-6D + A)x^4 + (4D - 8E + B)x^3 + (4D - 16E + 4F - 32D + C + 16E - 32D)x^2 + (-8D + 24E - 16F + 64D - 32E)x + (32D - 32E) = x^2 - 8x + 16[/tex]

Comparing the coefficients of like terms, we get the following system of equations:

-6D + A = 0

4D - 8E + B = 0

-24D + 4F - 32D + C = 16

-8D + 24E - 16F + 64D - 32E = 0

32D - 32E = 0

Solving this system of equations, we get:

D = E = 1

A = -6

B = -16

C = -80

F = -44

Therefore, the particular solution to the differential equation is:

[tex]y \_p(x) = (-6x^2 - 16x - 80) + e^{(2x)}(x^2 + x - 44)[/tex]

The general solution to the differential equation is the sum of the particular solution and the complementary function, which is the solution to the homogeneous equation:

[tex]y'''' - 2y'' + y = 0[/tex]

The characteristic equation of this homogeneous equation is:

[tex]r^4 - 2r^2 + 1 = 0[/tex]

Factoring the characteristic equation, we get:

[tex](r^2 - 1)^[/tex].

The particular solution to the differential equation by the method of undetermined coefficients is [tex]y \_p(x) = (-6x^2 - 16x - 80) + e^{(2x)}(x^2 + x - 44).[/tex]

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Answer:

Set your calculator to degree mode.

[tex] { \sin }^{ - 1} \frac{18}{20} = 64 [/tex]

So theta measures approximately 64 degrees.

Answer:

[tex] {6}^{2} + {b}^{2} = {10}^{2} [/tex]

[tex]36 + {b}^{2} = 100[/tex]

[tex] {b}^{2} = 64[/tex]

[tex]b = 8[/tex]

From the given information, we know that {an} is a sequence of positive terms, so all of its terms are greater than 0. We also know that sn = m∑ ak, which means that sn is a sum of a finite number of positive terms.

Now, let's look at the given limit: lim an+1 = 0 as n approaches infinity. This tells us that the terms of {an} must approach 0 as n approaches infinity since the limit of an+1 is dependent on the limit of an. Therefore, we can conclude that {an} is a decreasing sequence of positive terms. Using this information, we can determine the following:- ak must converge: Since {an} is decreasing and positive, we know that the terms of {ak} are also decreasing and positive. Therefore, {ak} must converge by the Monotone Convergence Theorem. - ak must converge to 0: Since {an} approaches 0 as n approaches infinity, we know that the terms of {ak} must also approach 0. Therefore, {ak} must converge to 0.
- {sn} must be bounded: Since {ak} converges to 0, we know that there exists some N such that ak < 1 for all n > N. Therefore, sn < m(N-1) + m for all n > N. This shows that {sn} is bounded above by some constant.
- {sn} is monotonic: Since {an} is decreasing and positive, we know that {ak} is also decreasing and positive. Therefore, sn+1 = sn + ak+1 < sn, which shows that {sn} is a decreasing sequence. - limn→∞ sn does not exist: Since {an} approaches 0 as n approaches infinity, we know that {sn} approaches a finite limit if and only if {ak} approaches a nonzero limit. However, we know that {ak} approaches 0, so {sn} does not approach a finite
Therefore, the correct answers
- ak must converge
- ak must converge to 0
- {sn} must be bounded
- {sn} is monotonic
- limn→∞ sn does not exist

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The limit of the given expression as x approaches 7 is 1/14.

To evaluate the limit:

lim x → 7 (x - 7) / ([tex]x^2[/tex] - 49)

We can see that this is an indeterminate form of type 0/0, since both the numerator and denominator approach 0 as x approaches 7. We can use L'Hospital's rule to evaluate this limit:

lim x → 7 (x - 7) / ([tex]x^2[/tex] - 49)

= lim x → 7 1 / (2x) [by applying L'Hospital's rule once]

= 1 / 14 [substituting x = 7]

Therefore, the limit of the given expression as x approaches 7 is 1/14.

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The derivatives are:

[tex]y' = (2x + 4y) / (4x - 6y)[/tex]

[tex]y'' = [(4x - 6y)(2 + 4((2x + 4y) / (4x - 6y))) - (2x + 4y)(4 - 6((2x + 4y) / (4x - 6y)))] / (4x - 6y)^2[/tex]

To find y' and y'' for the given equation x^2 + 4xy - 3y^2 = 8, follow these steps:

Step 1: Differentiate both sides of the equation with respect to x.
For the left side, use the product rule for 4xy and the chain rule for -3y^2.
[tex]d(x^2)/dx + d(4xy)/dx - d(3y^2)/dx = d(8)/dx[/tex]

Step 2: Calculate the derivatives.
[tex]2x + 4(dy/dx * x + y) - 6y(dy/dx) = 0[/tex]

Step 3: Solve for y'.
Rearrange the equation to isolate dy/dx (y'):
[tex]y' = (2x + 4y) / (4x - 6y)[/tex]

Step 4: Differentiate y' with respect to x to find y''.
Use the quotient rule: [tex](v * du/dx - u * dv/dx) / v^2[/tex],

where u = (2x + 4y) and v = (4x - 6y).
[tex]y'' = [(4x - 6y)(2 + 4(dy/dx)) - (2x + 4y)(4 - 6(dy/dx))] / (4x - 6y)^2[/tex]

Step 5: Substitute y' back into the equation for y''.
[tex]y'' = [(4x - 6y)(2 + 4((2x + 4y) / (4x - 6y))) - (2x + 4y)(4 - 6((2x + 4y) / (4x - 6y)))] / (4x - 6y)^2[/tex]

This is the expression for y'' in terms of x and y.

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The area of the other octagon is 75 square inches.

To find the area of the other octagon, we can use the concept of scale factors. The scale factor of 4:5 tells us that corresponding lengths of the two similar octagons are in a ratio of 4:5.

Since the scale factor applies to the lengths, it will also apply to the areas of the two octagons. The area of a shape is proportional to the square of its corresponding side length.

Let's assume the area of the other octagon (with the scale factor of 4:5) is A.

The ratio of the areas of the two octagons can be expressed as:

(Area of the given octagon) : A = (Side length of the given octagon)^2 : (Side length of the other octagon)^2

48 : A = (4/5)^2

48 : A = 16/25

Cross-multiplying:

25 * 48 = 16A

1200 = 16A

Dividing both sides by 16:

75 = A

Therefore, the area of the other octagon is 75 square inches.

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To get the running time equation of the given program, let's analyse it step by step.

The program consists of the following operations:
Step:1. Check if the length of the list is less than 2.
Step:2. Divide the list into two parts (left and right).
Step:3. Iterate through the left part and calculate the sum.
Step:4. Call the function recursively on the left part.
The running time equation can be represented as T(n), where n is the length of the list. The steps can be analyzed as follows:
1. The comparison takes constant time, so O(1).
2. Dividing the list also takes constant time, O(1).
3. Iterating through the left part takes O(n/2) as it processes half of the list.
4. Recursively calling the function with half of the list will have a running time of T(n/2).
Putting everything together, we get the following equation: T(n) = T(n/2) + O(n/2) + O(1)
This represents the running time equation of the given program.

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The functions $y =[tex]e^{-4x}[/tex]$ and $y = [tex]e^{2x}[/tex] $ are solutions to the differential equation $y'' + 2y' - 8y = 0$.

To see if the function $y = e^{rx}$ is a solution to the differential equation $y'' + 2y' - 8y = 0$, we substitute it in place of $y$ and its derivatives:

y=[tex]e^{rx}[/tex]

y' = [tex]re^{rx}[/tex]

y" = [tex]r^{2} e^{rx}[/tex]

Substituting these expressions into the differential equation, we get:

[tex]r^{2} e^{rx} + 2re^{rx} - 8e^{rx} = 0[/tex]

Dividing both sides by $ [tex]$e^{rx}$[/tex] $, we get:

[tex]r^{2} + 2r - 8 = 0[/tex]

This is a quadratic equation in $r$. Solving for $r$, we get:

r = -4,2

Therefore, the functions $y =[tex]e^{-4x}[/tex]$ and $y = [tex]e^{2x}[/tex] $ are solutions to the differential equation $y'' + 2y' - 8y = 0$.

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Okay, let's think this through step-by-step:

* A to D is 1 km

* B to C is 2 km

* B to D is 3 km

* A to B is 4 km

* C to D is 5 km

So we have:

A -> B = 4 km

B -> C = 2 km

C -> D = 5 km

We want to find A -> C.

A -> B is 4 km

B -> C is 2 km

So A -> C = 4 + 2 = 6 km

Therefore, the distance between stops A and C is 6 km.

Using the laws of simplification of fractions, we can find that in the simplest terms, cos M has a fraction value of 3/5.

In order to express a piece of a whole or a ratio of two numbers, a fraction requires a numerator (top number) and a denominator (bottom number) separated by a fraction bar.

The ratio of the neighbouring side to the hypotenuse of a right triangle is known as the cosine of an angle.

As a result, to calculate cos M, we must find the side that is perpendicular to M and divide it by the hypotenuse.

The length of the triangle's third side, KL, can be calculated using the Pythagorean theorem as shown below:

KL² + LM² = KM²

12² + 9² = 15²

144 + 81 = 225

225 = 15²

Taking the square root of both sides:

KL = √ (15² - 12²)

KL = √ (225 - 144)

KL = √81

KL = 9

As a result, angle M's neighbouring side, KL, has a length of 9. Therefore, by dividing 9 by 15, we can calculate cos M:

KL/KM = cos M = 9/15

To make this fraction simpler, divide the numerator and denominator by their 3 largest common factor:

cos M = (9/3)/ (15/3) = 3/5

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In a binary tree with leaves l1, l2, ..., lM at depths d1, d2, ..., dM respectively, the sum of [tex]2^-^d^_i[/tex] for all leaves is always less than or equal to 1: Σ  [tex]2^-^d^_i[/tex] <= 1.

In a binary tree, each leaf node is reached by following a unique path from the root. Since it is a binary tree, each internal node has two child nodes.

Consider a full binary tree, where all leaves have the maximum number of nodes at each depth. For a full binary tree, the total number of leaves is  [tex]2^d[/tex] , where d is the depth.

Each leaf node contributes [tex]2^-^d[/tex] to the sum. Thus, the sum for a full binary tree is Σ  [tex]2^-^d[/tex] = (2⁰ + 2⁰ + ... + 2⁰) = [tex]2^d[/tex] * [tex]2^-^d[/tex]  = 1. Now, if we remove any node from the full binary tree, the sum can only decrease, as we are reducing the number of terms in the sum. Hence, for any binary tree, the sum Σ [tex]2^-^d^_i[/tex]  will always be less than or equal to 1.

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