Find the area between the two curves on [2,6].y = e^xy = (1/x)

The value of f(t) is tan(t)  + sec(t)  + c.

What is integration?

Calculating areas, volumes, and their extensions requires the use of integrals, which are the continuous equivalent of sums. One of the two basic operations in calculus, along with differentiation, is integration, which is the process of computing an integral.

Here, we have

Given: f'(t) = sec(t) (sec(t) + tan(t))....(1)

We have to find the value f(t).

We take the integral of equation(1) and we get

∫f'(t) = ∫sec²(t)dt + ∫sec(t)tan(t)dt

We let

u = sec(t)

sec(t)tan(t)dt = du

∵ ∫sec²(x)dx = tan(x) + c

∫f'(t) = tan(t) + ∫1 du

f(t)  = tan(t)  + u + c

We substitute the value of u =  sec(t)

f(t) =  tan(t)  + sec(t)  + c

Hence, the value of f(t) is tan(t)  + sec(t)  + c.

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Answer:

B

Step-by-step explanation:

It has to reflect from one side to the other therefor left to right.

Answer:

B

Step-by-step explanation:

Both of them are congruent, meaning that they are the same but are reflected.

a. To find the z-score such that 48% of the probability is within z standard deviations of the mean, we need to find the z-score such that the area to the right of z is (1-0.48)/2 = 0.26. Using a standard normal distribution table or a calculator, we find that this corresponds to a z-score of approximately 0.68 (rounded to two decimal places).

b. To find the z-score such that 81% of the probability is within z standard deviations of the mean, we need to find the z-score such that the area to the right of z is (1-0.81)/2 = 0.095. Using a standard normal distribution table or a calculator, we find that this corresponds to a z-score of approximately 1.41 (rounded to two decimal places).

c. Below is a sketch of the standard normal distribution with the area within one and two standard deviations of the mean shaded. The z-scores corresponding to these areas are approximately -1 and 1, respectively. To find the area within 0.68 standard deviations of the mean (corresponding to part a), we would shade the area between -0.68 and 0.68. To find the area within 1.41 standard deviations of the mean (corresponding to part b), we would shade the area between -1.41 and 1.41.

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The numeric value of the composition of the functions is given as follows:

f ( g ( f ( f ( f ( g ( 27 ) ) ) ) ) ) = 1.

To calculate the numeric value of a function or of an expression, we substitute each instance of any variable or unknown on the function by the value at which we want to find the numeric value of the function or of the expression presented in the context of a problem.

The functions for this problem are given as follows:

We obtain the numeric values from the inside out, hence:

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The matrix which represents R with respect to standard coordinates is -

[tex]\left[\begin{array}{ccc}cos(17)^{o} &sin(17)^{o}&0\\-sin(17)^{o}&cos(17)^{o}&0\\0&0&1\end{array}\right][/tex]

Given is that R : R → R* be the rotation with the properties. The axis of rotation is the line L, spanned and oriented by the vector v = (3,-1,3). R is rotated about L through the angle t = 17 according to the Right Hand Rule

We have θ = 17°.

The given cartesian vector is -

3i - j + 3k

We can write the matrix as -

[tex]\left[\begin{array}{ccc}cos(17)^{o} &sin(17)^{o}&0\\-sin(17)^{o}&cos(17)^{o}&0\\0&0&1\end{array}\right][/tex]

So, the matrix which represents R with respect to standard coordinates is -

[tex]\left[\begin{array}{ccc}cos(17)^{o} &sin(17)^{o}&0\\-sin(17)^{o}&cos(17)^{o}&0\\0&0&1\end{array}\right][/tex]

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, A = 3, B = 4, C = 6, and D = 1. Therefore, the equation of the hyperbola is:

[tex](x - 6)^2 / 9 - (y - 1)^2 / 16 = 1[/tex]

To find the equation of the hyperbola with these given parameters, we can use the standard form equation:

[tex](x - h)^2 / a^2 - (y - k)^2 / b^2 = 1[/tex]

where (h, k) is the center of the hyperbola, a is the distance from the center to the vertex/foci, and b is the distance from the center to the asymptotes.

From the given information, we know that the center is (6, 1), the focus is (11, 1), and the vertex is (9, 1). We can use the distance formula to find a and c (the distance from the center to the foci):

a = distance from (6, 1) to (9, 1) = 3
c = distance from (6, 1) to (11, 1) = 5

Using the formula[tex]c^2 = a^2 + b^2,[/tex]we can solve for b:

[tex]25 = 9 + b^2[/tex]
[tex]b^2 = 16[/tex]
b = 4

Now we have all the values we need to plug into the standard form equation:

[tex](x - 6)^2 / 9 - (y - 1)^2 / 16 = 1[/tex]

To write this in the form (x - C)^2 / A^2 - (y - D)^2 / B^2 = 1, we can rearrange the terms and write:

[tex](x - 6)^2 / 3^2 - (y - 1)^2 / 4^2 = 1[/tex]

So, A = 3, B = 4, C = 6, and D = 1. Therefore, the equation of the hyperbola is:

[tex](x - 6)^2 / 9 - (y - 1)^2 / 16 = 1[/tex]
And in the form[tex](x - C)^2 / A^2 - (y - D)^2 / B^2 = 1,[/tex] it is:

[tex](x - 6)^2 / 3^2 - (y - 1)^2 / 4^2 = 1[/tex]

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To find the time t when the instantaneous velocity of the ball equals its average velocity, we need to use the formula for average velocity:
average velocity = (change in position) / (change in time)
We can also find the instantaneous velocity by taking the derivative of the position function s(t):
instantaneous velocity = s'(t)
Let's find the average velocity over a certain time interval. Let's say we want to find the average velocity over the interval from t = 0 to t = 5 seconds. Then the change in position would be:
change in position = s(5) - s(0) = (-10 + 250) - (-10 + 250) = 0
And the change in time would be:
change in time = 5 - 0 = 5 seconds
So the average velocity over this time interval is:
average velocity = 0 / 5 = 0 ft/s
Now let's find the instantaneous velocity at time t. Taking the derivative of s(t), we get:
s'(t) = -10
So the instantaneous velocity is a constant -10 ft/s, regardless of the time t.
To find the time t when the instantaneous velocity equals the average velocity, we set these two equal to each other:
s'(t) = average velocity
-10 = 0
This equation has no solution, which means the instantaneous velocity never equals the average velocity. Therefore, there is no time t when this occurs.

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function A,b, and c are linear. shown below are the graph function A in standard (x,y) coordinate pane, a table of 5 ordered pairs belonging to function B, and an equation for function C. Arrange the functions in order of their rates of change from least to greates.

The dimensions of the box for which the volume is maximized are  2√15 cm and 5√15 cm.

Let's denote the side length of the square base by "x" and the height of the box by "h". Then, the total surface area of the box is:

S = [tex]x^{2}[/tex] + 4xh

We know that S = 300, so we can write:

[tex]x^{2}[/tex] + 4xh = 300

To maximize the volume of the box, we need to find the values of x and h that satisfy this equation and give us the largest possible value for V, the volume of the box.

The volume of the box is given by:

V = [tex]x^{2}[/tex]h

To find the maximum value of V, we can use the method of Lagrange multipliers. We want to maximize V subject to the constraint that S = 300, so we define the Lagrangian function:

L(x, h, λ) = [tex]x^{2}[/tex]h + λ([tex]x^{2}[/tex] + 4xh - 300)

To find the maximum of V, we need to solve the system of equations:

∂L/∂x = 2xh + 2λx + 4λh = 0

∂L/∂h = [tex]x^{2}[/tex] + 4λx = 0

∂L/∂λ = [tex]x^{2}[/tex] + 4xh - 300 = 0

Solving these equations, we get:

h = 5x/2

[tex]x^{2}[/tex] = 60

Substituting h = 5x/2 and [tex]x^{2}[/tex] = 60 into the equation for the volume, we get:

V = [tex]x^{2}[/tex]h = (60)(5x/2) = 150x

So, to maximize the volume, we need to find the value of x that maximizes V. Since [tex]x^{2}[/tex] = 60, we have x = √60 = 2√15. Substituting this value into the equation for h, we get:

h = 5x/2 = 5(2√15)/2 = 5√15

Therefore, the dimensions of the box for which the volume is maximized are:

length of the side of the square base = 2√15 cm

height of the box = 5√15 cm.

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Answer: The cost of each game increases the price by $4.

Step-by-step explanation: More

the surface area of the rectangular prism with a length of 2.1 cm, width of 1.81 cm, and height of 6 cm is 37.34 square centimeters.

What is a rectangle?

Rectangles are quadrilaterals having four right angles in the Euclidean plane of geometry. Various definitions include an equiangular quadrilateral, A closed, four-sided rectangle is a two-dimensional shape. A rectangle's opposite sides are equal and parallel to one another, and all of its angles are exactly 90 degrees.

We are given the formula for surface area of a rectangular prism: SA = 2(w) + 2(wh) + 2(h).

Substituting the given values, we get:

SA = 2(1.81) + 2(1.81 × 6) + 2(6)

SA = 3.62 + 21.72 + 12

SA = 37.34

Therefore, the surface area of the rectangular prism with a length of 2.1 cm, width of 1.81 cm, and height of 6 cm is 37.34 square centimeters.

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The  natural annual percent of  increase of its population is 1.6%, under the given condition that in the given country possess a crude birth rate of 24 per 1,000 and crude death rate of 8 per 1,000.

Then the correct option is Option B.

For the purpose of evaluating the natural annual percent of increase in a population we have to  subtract crude death rate from  crude birth rate and then dividing by 10.

So for the given case,

the natural annual percent of increase in the population would be

((24-8)/10)

= 1.6%

The  natural annual percent of  increase of its population is 1.6%, under the given condition that in the given country possess a crude birth rate of 24 per 1,000 and crude death rate of 8 per 1,000.



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The complete question

If a country has a crude birth rate of 24 per 1,000 and a crude death rate of 8 per 1,000, the natural annual percent increase of its population is

a) 0.6%

b)1.6%

c) 3%

d)16%

e) 32%

The set of side lengths that form a right triangle are: .15 m, 20 m, 25 m.

A fundamental rule of geometry known as the Pythagorean theorem asserts that the square of the length of the hypotenuse, the longest side in a right triangle, is equal to the sum of the squares of the lengths of the other two sides.

When the lengths of the other two sides of a right triangle are known, the Pythagorean theorem is used to determine the length of the third side. It is also used to determine whether a set of three side lengths, like in the previous question, constitutes a right triangle. In mathematics, physics, and engineering, the Pythagorean theorem is used to solve a variety of vector and force-related problems as well as to compute distances, areas, and volumes.

The Pythagoras Theorem is given as:

a² + b² = c²

For the given values of side lengths we have:

a. 10² = 7² + 8² No true

b. 25² = 15² + 20². True

c. 10² + 40² = 41² Not true

d. 5² = 3² + 6². Not True

Hence. the side lengths that form a right triangle are: .15 m, 20 m, 25 m.

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The probability distribution for the proportion of the 206 polled voters that support a new tax on fast food can be modeled by a normal distribution with a mean of 0.40 and a standard deviation of 0.0341 (rounded to four decimal places).

To answer your question, we need to find the mean and standard deviation for the normal probability distribution representing the proportion of polled voters that support a new tax on fast food.

1. Calculate the mean (µ):
The mean of the proportion can be found using the formula µ = p, where p is the proportion of voters that support the tax. In this case, p = 0.40. So, µ = 0.40.

2. Calculate the standard deviation (σ):
The standard deviation for a proportion can be calculated using the formula σ = √[p(1-p)/n], where n is the number of voters polled. In this case, n = 206.
σ = √[0.40(1-0.40)/206] = √[0.24/206] = √0.001165 = 0.0341 (rounded to 4 decimal places)

3. Complete the boxes with mean and standard deviation values:
Mean (µ): 0.40
Standard Deviation (σ): 0.0341

The probability distribution for the proportion of the 206 polled voters that support a new tax on fast food can be modeled by a normal distribution with a mean of 0.40 and a standard deviation of 0.0341 (rounded to four decimal places).

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the vectors u and v are neither parallel nor orthogonal.

What is Orthogonal?

In mathematics, two vectors are said to be orthogonal if they are perpendicular to each other, which means that they meet at a right angle. More generally, in a Euclidean space of any number of dimensions, two vectors are orthogonal if their dot product is zero. This means that the cosine of the angle between them is zero, which implies that the angle between them is 90 degrees (or pi/2 radians). Orthogonal vectors are important in various areas of mathematics, including linear algebra, calculus, and geometry.

The magnitudes of u and v can be found using the Pythagorean theorem:

[tex]||u|| = \sqrt{((-7)^2 + (-4)^2)} = \sqrt{(49 + 16)} = \sqrt{(65)}\\v = \sqrt{((-16)^2 + 28^2)} = \sqrt{(256 + 784)} = \sqrt{(1040)}[/tex]

Now we can calculate the dot product of u and v:

u · v = (-7)(-16) + (-4)(28) = 112

Putting it all together, we get:

[tex]112 = \sqrt{65} \sqrt{1040} cos(\theta)\\cos(\theta) = 112 / (\sqrt{65} \sqrt{1040})\\cos(\theta) = 0.926[/tex]

Since the cosine of the angle θ is positive and greater than zero, we can conclude that the angle is acute and the vectors u and v are not orthogonal.

So, in summary, the vectors u and v are neither parallel nor orthogonal.

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Answer:

The vectors are orthogonal because u⋅v = 0.

Step-by-step explanation:

o7

The integral [tex]∫59/sqrt(1-x^2)[/tex] dx is convergent, and its value is [tex]59π/a^2[/tex]. The substitution x = a sin(t) was used to simplify the integral, and the limits of integration were transformed from -1 to 1 to -π/2 to π/2.

The integral[tex]∫59/sqrt(1-x^2)[/tex] dx is a definite integral that represents the area under the curve of the function[tex]59/sqrt(1-x^2)[/tex] between its limits of integration. To determine whether this integral is convergent or divergent, we need to evaluate the integral by using a suitable technique.

We can begin by noting that the integrand is of the form [tex]f(x) = k/√(a^2-x^2)[/tex], where k and a are constants. This suggests that we should use the substitution x = a sin(t) to simplify the integral.

Making this substitution, we obtain dx = a cos(t) dt and the limits of integration become -π/2 to π/2. The integral now becomes:

[tex]∫59/sqrt(1-x^2) dx = ∫59/(a cos(t)) a cos(t) dt[/tex][tex]= 59∫(1/a^2) dt = 59t/a^2[/tex]

Evaluating the integral from -π/2 to π/2, we obtain:

[tex]∫59/sqrt(1-x^2) dx[/tex][tex]= 59(π/2 - (-π/2))/a^2[/tex][tex]= 59π/a^2[/tex]

Since the limits of integration are finite, and the integral has a finite value, we can conclude that the integral is convergent. Evaluating the integral using the substitution x = a sin(t), we obtain the value[tex]59π/a^2[/tex].

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The value of f(x) is [tex]f(x) =\frac{2}{5}x^5+\frac{5}{2}x^{2} +x-\frac{39}{10}[/tex]

The equation in which the derivative of the given function is included is known as the differential equation. We have to find out a particular solution to the given ODE. We will use the power rule of integration to solve this question.

We have the function :

f"(x) = [tex]8x^3+5[/tex]

Integrate on both sides with respect to x.

[tex]f'(x) = 8\int\limits x^3dx + \int\limits 5dx\\\\f'(x) = 2x^4+5x+C_1[/tex]

Integrate on both sides with respect to x.

[tex]f(x) = 2\int\limitsx^4dx+5\int\limits xdx+\int\limits C_1dx\\\\f(x) = \frac{2}{5}x^5+\frac{5}{2}x^2+C_1x+C_2\\ \\[/tex]

f'(1) = 8

 8 = 2 + 5 + [tex]C_1[/tex]

[tex]C_1=0[/tex]

f(1) =0

[tex]0 = \frac{2}{5} +\frac{5}{2}+1+C_2\\ \\C_2=-\frac{39}{10\\}\\[/tex]

[tex]f(x) =\frac{2}{5}x^5+\frac{5}{2}x^{2} +x-\frac{39}{10}[/tex]

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Answer:

977.44

Step-by-step explanation:

divide by 9

you get 977.44444

leave your answer to 2 decimal places

The values of t for which the curve has a vertical tangent are t=1 and t=-1. Note that t=4/9 is not one of these values, so the given information about t=4/9 is not relevant to this question.

To determine all values of t for which the given curve has a vertical tangent, we need to find the values of t where the derivative of y with respect to x (dy/dx) is undefined (i.e., where the slope of the tangent line is vertical or infinite).

Using the chain rule, we can find that:

dy/dx = (dy/dt)/(dx/dt) = (6t - 8t)/(6t^2 - 6) = -2(t-4)/(t^2-1)

To have a vertical tangent, we need dy/dx to be undefined, which means the denominator (t^2-1) must be equal to zero. Therefore, we have:

t^2 - 1 = 0
(t-1)(t+1) = 0
t = 1 or t = -1

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The time until the minimum price is reached is D)9.6 months

To find the time until the minimum price is reached, we need to find the value of t that minimizes the function P(t) = 130t^2 - 2500t + 6900.

One way to do this is to take the derivative of P(t) with respect to t, and set it equal to zero to find the critical point(s):

P'(t) = 260t - 2500 = 0

t = 2500/260 = 9.6 months

So the critical point is at t = 9.6 months. To check that this is a minimum, we can take the second derivative of P(t):

P''(t) = 260

Since P''(t) is positive for all t, we know that the critical point at t = 9.6 months is a minimum.

Therefore, the answer is D) 9.6 months.

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Therefore, the probability that more than one drug in a batch is defective is 0.347.

To solve this problem, we can use the binomial probability distribution. Let X be the number of defective drugs in a batch of 15. Then, X follows a binomial distribution with parameters n = 15 and p = 0.08.
(i) To determine the probability that none of the drugs in a batch is defective, we need to find P(X = 0). This can be calculated using the binomial probability formula:
P(X = 0) = (15 choose 0) × [tex]0.08^0[/tex] × [tex]0.92^{15}[/tex] = 0.327
Therefore, the probability that none of the drugs in a batch is defective is 0.327.
(ii) To determine the probability that more than one drug in a batch is defective, we need to find P(X > 1). This can be calculated using the binomial probability formula and some algebra:
P(X > 1) = 1 - P(X <= 1)
         = 1 - P(X = 0) - P(X = 1)
         = 1 - [(15 choose 0) × [tex]0.08^0[/tex] × [tex]0.92^{15}[/tex] + (15 choose 1) × [tex]0.08^1[/tex] × [tex]0.92^{14}[/tex]]
         = 0.347

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The interval of convergence is:
I = (-1, 1)
So, the radius of convergence is:
R = 1

To find the radius of convergence and interval of convergence of the given series, we'll use the Ratio Test. The given series is:

Σ (from n=1 to infinity) 5(-1)^n nx^n

Let's consider the absolute value of the general term and apply the Ratio Test:

L = lim (n -> infinity) | (5(-1)^(n+1) (n+1)x^(n+1)) / (5(-1)^n nx^n) |

L = lim (n -> infinity) | ((-1)(n+1)x) / n |

Now, let's find the limit:

L = |-x| lim (n -> infinity) | (n+1) / n |

The limit is 1 as n goes to infinity. Therefore:

L = |-x|

For the Ratio Test, if L < 1, the series converges. So:

|-x| < 1

This inequality gives us the interval of convergence:

-1 < x < 1

Thus, the interval of convergence is:

I = (-1, 1)

The radius of convergence (R) is the distance from the center of the interval to either endpoint:

R = (1 - (-1)) / 2 = 2 / 2 = 1

So, the radius of convergence is:

R = 1

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The convergence range is I = [-1/5, 1/5)

We employ the ratio test to determine the radius of convergence:

lim┬(n→∞)⁡|5(-1)nx| = lim(n)|x(n+1)/n| = lim(n)|(n+1)/n| = n (n+1)x(n+1)|/|5(-1)n nx|

As a result, R = 1/5 is the radius of convergence.

Test the endpoints x = -1/5 and x = 1/5 to determine the interval of convergence:      

The series changes to: when x = -1/5

Σ 5(-1)^n n(-1/5)^n = Σ (-1)^n n/5^n

Since n/5n is decreasing and this alternate series has diminishing terms, it converges according to the alternating series test. Therefore, the interval of convergence includes x = -1/5.

The series changes to: when x = 1/5.

Σ 5(-1)^n n(1/5)^n = Σ (n/5)^n

Since this series is positive, we can perform the ratio test:

lim┬(n→∞)⁡|(n+1)/5|^(n+1)/(n/5)"n" = lim(n)(n+1).^{n+1}/n^n/5 = ∞

When x = 1/5, the series diverges, according to the ratio test.

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Answer:

[tex]f'(x)= \frac{13}{(2x+3)^2}\\[/tex]

Step-by-step explanation:

[tex]f(x)= \frac{3x-2}{2x+3} \\[/tex]

[tex]f'(x)=\frac{dy}{dx} = \frac{d}{dx}(\frac{3x-2}{2x+3})\\ f'(x)= \frac{(2x+3)\frac{d}{dx}(3x-2)-(3x-2)\frac{d}{dx}(2x+3) }{(2x+3)^{2} } \\f'(x)= \frac{(2x+3)(3)-(3x-2)(2)}{(2x+3)^{2} } \\[/tex]

[tex]f'(x)= \frac{6x+9-6x+4}{(2x+3)^{2} }\\ f'(x)= \frac{13}{(2x+3)^2}\\[/tex]

When the population standard deviation is unknown and the sample size is less than 30, the t-table value should be used in computing a confidence interval for a mean. (option a. t.)

When the population standard deviation is unknown and the sample size is less than 30, the appropriate table value to use for computing a confidence interval for a mean is the t-distribution table. This is because the t-distribution is used when the sample size is small and the population standard deviation is unknown.

The t-distribution table gives critical values for a given level of confidence and degrees of freedom (df), where df is equal to the sample size minus one (df = n - 1). The critical value from the t-distribution table is used to calculate the margin of error for the confidence interval.

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To minimize the cost of laying the pipe, point P should be located approximately 2.7 km from point A.

1. Let x be the distance from A to P.
2. Use the Pythagorean theorem to find the distance from W to P: WP = √((5 km)² + x²).

3. The underwater distance is WP, and the overland distance is (8 - x) km.

4. Calculate the total cost: C = P10,000,000(WP) + P5,000,000(8 - x).

5. Differentiate C with respect to x: dC/dx = P10,000,000(1/2)(1/√(25 + x²)(2x)) - P5,000,000.

6. Set dC/dx = 0 to find the minimum cost: x ≈ 2.7 km.

7. Point P is approximately 2.7 km from point A along the shoreline.

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The absolute maximum value of f(t) on the interval [-1,5] is 3

To find the absolute maximum and absolute minimum values of the function f(t) = t - √(t-1) on the interval [-1,5], we need to first find the critical points and endpoints of the interval.

Taking the derivative of f(t), we get:

f'(t) = 1 - 1/2(t-1)[tex]^{(-1/2)[/tex]

Setting this equal to zero and solving for t, we get:

1 ([tex]\frac{-1}{2}[/tex]) (t-1)([tex]\frac{-1}{2}[/tex])  = 0

[tex]\frac{1}{2}[/tex](t-1)([tex]\frac{-1}{2}[/tex])  = 1

(t-1)([tex]\frac{-1}{2}[/tex]) = 2

t-1 = 1/4

t = 1.25

The critical point is at t = 1.25.

Now, we need to check the function at the endpoints and the critical point to determine the absolute maximum and absolute minimum values.

f(-1) = -1 - √(-1-1) = -2

f(5) = 5 - √(5-1) = 3

f(1.25) = 1.25 - √(1.25-1) ≈ 0.354

Therefore, the absolute maximum value of f(t) on the interval [-1,5] is 3, which occurs at t=5, and the absolute minimum value of f(t) on the interval is approximately 0.354, which occurs at t = 1.25.

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The acceleration of the rocket is 74 m/s².

The rate at which an object changes its velocity over time is called acceleration.

It has both magnitude (the change in velocity) and direction because it is a vector quantity.

In simpler terms, acceleration is the rate at which an object changes direction or speed.

Assuming an article speeds up, dials back, or takes a different path, it can speed up.

In the metric system, acceleration is typically measured in meters per second squared (m/s²), whereas in the imperial system, acceleration is measured in feet per second squared (ft/s²).

To determine the acceleration of the rocket, we need to use Newton's Second Law of Motion, which states that the acceleration of an object is directly proportional to the net force, inversely proportional to its mass, acting on it.

Mathematically, this can be expressed as:

a = F_net / m

In this case, the net force acting on the rocket is the force generated by the engine, which is 7.4 N. The mass of the rocket is 0.1 kg. Therefore, we can plug in these values into the formula above and get:

a = 7.4 N / 0.1 kg

a = 74 m/s²

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The only solution that satisfies the given conditions is r = 0, y = 2, and Q = r'y = 0.

To find the two positive numbers r and y that maximize Q=r'y, we need to use the Lagrange multiplier method. Let's define a Lagrangian function L(r, y, λ) as follows:
L(r, y, λ) = r'y + λ(x + y - 2)
where λ is the Lagrange multiplier. We need to find the values of r, y, and λ that maximize L(r, y, λ).
Taking partial derivatives of L with respect to r, y, and λ, we get:
∂L/∂r = y
∂L/∂y = r + λ
∂L/∂λ = x + y - 2
Setting these partial derivatives equal to zero, we get:
y = 0 (this is not a valid solution as we need positive numbers)
r + λ = 0
x + y - 2 = 0
From the second equation, we get r = -λ. Substituting this into the first equation, we get y = 0. Substituting r and y into the third equation, we get x = 2.

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The value of the integral is 9. To evaluate the integral: ∫[1,8] [tex]x^{(-2/3)}[/tex] dx

We can use the power rule of integration. Specifically, we have:

∫ [tex]x^{(-2/3)}[/tex] dx = 3[tex]x^{(1/3)}[/tex] / (1/3) + C = 9[tex]x^{(1/3)}[/tex] + C

where C is the constant of integration.

Applying this formula to the given integral, we have:

∫[1,8] [tex]x^{(-2/3)}[/tex] dx = [9x^(1/3)] [1,8] = 9([tex]8^{(1/3)}[/tex] - [tex]1^{(1/3)}[/tex]= 9(2 - 1) = 9

Therefore, the value of the integral is 9.

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The particular solution of the indicated linear system that satisfies the initial conditions is (1/5) e⁻ᵃ [1 1/5] + (8/5) t e⁻ᵃ [1 1/5] + (4/5) e⁻ᵃ [1 1/5] + (3/5) e²ᵃ [1 1]

The first step in finding a particular solution of a linear system that satisfies given initial conditions is to write the system in matrix form, which is already given as:

X ′ = [ 3 − 1

5 − 3]x

Here, X ′ is the derivative of the vector X with respect to time t, and x is the vector of unknown functions that we want to find. To solve this system, we need to find the eigenvalues and eigenvectors of the matrix [3 -1; 5 -3], which can be done by finding the roots of the characteristic equation det([3 -1; 5 -3] - λI) = 0, where I is the identity matrix and λ is the eigenvalue.

Solving the characteristic equation, we get λ = -1 and λ = -1, which means that we have one repeated eigenvalue. To find the eigenvectors, we need to solve the equation ([3 -1; 5 -3] - (-1)I)x = 0 for each eigenvalue. For λ = -1, we get the equation

[4 -1; 5 -2]x = 0

which has the general solution x = c[1; 1/5], where c is a constant. For a repeated eigenvalue, we also need to find the generalized eigenvectors, which are solutions of the equation ([3 -1; 5 -3] - (-1)I)x = v, where v is a nonzero vector orthogonal to the eigenvector.

For λ = -1, we can choose v = [0; 1] and solve the equation ([3 -1; 5 -3] - (-1)I)x = [0; 1], which gives the solution x = [1/5; 1/25]. Thus, the eigenvector matrix P and the generalized eigenvector matrix Q are

P = [1 1/5; 1 1/5] and Q = [1 1/5; 0 1/25]

respectively. Using these matrices, we can write the general solution of the system as

x = c₁ e⁻ᵃ [1 1/5] + c₂ t e⁻ᵃ [1 1/5] + c₃ e⁻ᵃ [1 1/5] + c4 e^(2t) [1 1]

where c₁, c₂, c₃, and c4 are constants determined by the initial conditions.

Now, we can use the given initial conditions x(0) = [1 1] to find the values of c₁, c₂, c₃, and c4. Substituting t = 0 and x = [1 1] into the general solution, we get

[1 1] = c₁ [1 1/5] + c₂ (0) [1 1/5] + c₃ [1 1/5] + c4 [1 1]

which simplifies to

c₁ + c₃ + c4 = 1

c₁ + (1/5)c₃ + c4 = 1

Using the given initial conditions x'(0) = [2 4], we can also find the values of c₂ and c₃ by differentiating the general solution and substituting t = 0 and x' = [2 4]. This gives us the equations

x'(0) = [-1 0]c₁ + [-1/5 + 1]c₂ + [-1/5]c₃ + [2 2]c4 = [2 4]

Simplifying this equation, we get

c₁ - (1/5)c₃ + 2c4 = 2

c₂ + 2c4 = 4

We now have a system of four equations in four unknowns, which can be solved using algebraic manipulation. Solving for c₁, c₂, c₃, and c4, we get

c₁ = 1/5

c₂ = 8/5

c₃ = 4/5

c4 = 3/5

Substituting these values back into the general solution, we get the particular solution that satisfies the given initial conditions:

x = (1/5) e⁻ᵃ [1 1/5] + (8/5) t e⁻ᵃ [1 1/5] + (4/5) e⁻ᵃ [1 1/5] + (3/5) e²ᵃ [1 1]

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Complete Question:

Find a particular solution of the indicated linear system that satisfies the given initial conditions.

X ′ = [ 3 − 1

          5 − 3]x

x_1 = e^(2t) [1    1]

x_2 = e^(-2t) [ 1    5]