Average Cost for Producing Microwaves Let the total cost function C(x) be defined as follows. C(x)= 0.0002-0.05x²+104x + 3,400 Find the average cost function C. (((x)= Find the marginal average cost function C C'(x)= Need Help? MY NOTES PRACTI

To evaluate the definite integral ∫(2e^(1/x^5) + 1) dx over the given interval, we'll start by finding the antiderivative of the integrand.

∫(2e^(1/x^5) + 1) dx = 2∫e^(1/x^5) dx + ∫1 dx

For the first term, we need to evaluate the integral of e^(1/x^5). This integral does not have a simple closed-form solution, so we'll leave it as is. Let's denote the integral of e^(1/x^5) as F(x):

∫e^(1/x^5) dx = F(x)

The integral of 1 with respect to x is simply x:

∫1 dx = x

Now, to evaluate the definite integral over a given interval, we subtract the values of the antiderivative at the upper and lower bounds:

∫[a, b] (2e^(1/x^5) + 1) dx = 2F(x) + x ∣[a, b]

Therefore, the evaluation of the definite integral depends on the specific values of the interval [a, b]. Please provide the values for a and b, and I can provide the numerical result.

Regarding the second definite integral, ∫[-π/16, π/16] cos(8x)sin(sin(8x)) dx, we'll follow a similar approach. Let's denote the integrand as f(x):

f(x) = cos(8x)sin(sin(8x))

To evaluate this integral, we'll employ numerical methods or a calculator since the integrand involves trigonometric functions and their composition.

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Answer:

10/33

Step-by-step explanation:

2 red discs + 5 pink discs + 4 purple discs = 11 discs

The chance of pulling a pink disc is 5/11

On a 6 sided dice, there are 6 numbers, 4 are greatee than 2. So the chance of rolling a number greater than 2 is 4/6.

5/11 × 4/6 = 20/66 = 10/33

The compound interest is 10% for the growth of principal amount to stated amount.

The correct compound interest formula to be used for the calculation is -

A = [tex] {p(1 + \frac{r}{n} )}^{nt} [/tex], where A is the amount, p is the principal, r is rate of interest, n is number of times the interest in compounded and t is time.

Since the interest is to be compounded semiannually, the n will be 2.

Keep the values in formula to find the value of interest rate.

75000 = [tex] {5000 (1 + \frac{r}{2} )}^{2×4} [/tex]

Divide the value and perform multiplication in power

75000/5000 = [tex] { (1 + \frac{r}{2} )}^{8} [/tex]

1.5 = [tex] { (1 + \frac{r}{2} )}^{8} [/tex]

Rearranging the equation

(1 + r/2) = [tex] { (1.5}^{1/8} [/tex]

r/2 = ([tex] { (1.5}^{1/8} [/tex] - 1)

Multiplying both sides with 2

r = 2 × ([tex] { (1.5}^{1/8} [/tex] - 1)

r = 2 × 0.05

Multiplying the values

r = 0.1 or 10%

Hence, the rate of interest is 10%.

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The complete question is -

Find the interest rate needed for an investment of $5000 to grow to an amount of $75000 in 4 years if interest is compounded semiannually. Use A = p 1+- - P(1+2) compound interest formula.

To find f^(-1)(a), we need more information about the function f and its inverse. Without that information, we cannot determine the values of C, D, and k in the form f^(-1)(x) = Cx + D, x ≠ k.

The question asks for the inverse of a function f, denoted as f^(-1)(a). However, without knowing the specific form or properties of the function f, we cannot determine its inverse or the values of C, D, and k in the form f^(-1)(x) = Cx + D, x ≠ k.

The notation (h, k) typically refers to the coordinates of a point or a hole in a function, but without any information about the function f, we cannot determine the coordinates (h, k) or their relation to the inverse function.

Therefore, without further details about the function f and its properties, it is not possible to provide the values of C, D, and k or determine the form of the inverse function f^(-1)(x).

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The common difference of the arithmetic sequence is 2. Let's use the formulas for the nth term and the sum of the first n terms of an arithmetic sequence to solve this problem.

The nth term of an arithmetic sequence is given by:

an = a1 + (n-1)d

where a1 is the first term, d is the common difference, and n is the number of the term. The sum of the first n terms of an arithmetic sequence is given by: Sn = n/2(2a1 + (n-1)d)

where a1 is the first term, d is the common difference, and n is the number of terms. We are given that the eleventh term of the arithmetic sequence is 30, so we can write: a11 = a1 + (11-1)d = a1 + 10d = 30

We are also given that the sum of the first eleven terms is 55, so we can write: S11 = 11/2(2a1 + 10d) = 55

Simplifying the equation for S11, we get: 2a1 + 10d = 10

We can now use the equation for a11 to eliminate a1 from the equation for S11: a11 = a1 + 10d = 30

a1 = 30 - 10d

Substituting into the equation for S11, we get: 2(30 - 10d) + 10d = 10

Simplifying and solving for d, we get: d = 2

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a. The distance between points A and C is  37.42 km.

b.  The bearing between point C and point A is found to be 71.48°.

a)

Segment 1: From point A to point B:

The ship sails a distance of 20 km on a bearing of 030°,

East displacement = 20 km * cos(30°)

= 20 km * (√3/2)

= 10√3 km

Segment 2: From point B to point C:

The ship sails a distance of 25 km on a bearing of 75°.

East displacement = 25 km * cos(75°)

= 25 km * (√3 + 1)/2

=  21.65 km

The total east displacement from point A to C :

Total east displacement = 10√3 km + 21.65 km

= 31.65 km

Distance between A and C = √(20 km)² + (Total east displacement)²

= √20² + (1.65²

≈ √400² + 1000.4225²

≈ √1400.4225 km²

=  37.42 km

b)

North displacement = (20 km * sin(30°)) + (25 km * sin(75°))

= 20 km * (1/2) + 25 km * (√3/2)

= 10 km + (25√3/2)

= 10 km + 21.65 km

=  31.65 km

Bearing of C with respect to A = arctan(31.65 km / Total east displacement)

= arctan(31.65 km / 10√3 km)

= 71.48°

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In a connected planar graph with 30 vertices, where each vertex has a degree of 5, the number of faces is 47.

According to Euler's formula for a planar graph, we have F = E - V + 2, where F represents the number of faces, E represents the number of edges, and V represents the number of vertices.

Given:

Number of vertices (V) = 30

Degree of each vertex = 5

To calculate the number of edges (E), we can use the fact that each vertex contributes an edge to the total count equal to its degree. Since each vertex has a degree of 5, the total number of edges is (V * 5) / 2, as each edge is counted twice.

E = (30 * 5) / 2 = 75

Now we can substitute the values into Euler's formula to find the number of faces (F):

F = E - V + 2

F = 75 - 30 + 2

F = 47

Therefore, the number of faces in the connected planar graph G is 47.

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the matrix A has three distinct eigenvalues and we were able to find three linearly independent eigenvectors, the matrix A is diagonalizable according to Theorem 7.5

ToTo find the eigenvalues and eigenvectors of matrix A, we solve the equation (A - λI)v = 0, where λ is the eigenvalue and v is the corresponding eigenvector.

The characteristic equation is det(A - λI) = 0.

Expanding the determinant, we get:
(2-λ)(3-λ)(0-λ) + (-1)(-2)(-1-λ) + (1)(-2)(-1-λ) - (1)(-2)(3-λ) - (1)(-1)(0-λ) - (-1)(-2)(1-λ) = 0

Simplifying the equation, we obtain:
-λ³ + 3λ² + 2λ - 4 = 0

Solving this equation, we find the eigenvalues:
λ = -1, 2, 2

To find the corresponding eigenvectors, we substitute each eigenvalue back into the equation (A - λI)v = 0 and solve for v.

For λ = -1:
(A + I)v = 0
[3 -1 1; -2 4 -2; -1 -1 1]v = 0

Solving the system of equations, we find v₁ = [1; 1; -1].

For λ = 2:
(A - 2I)v = 0
[0 -1 1; -2 1 -2; -1 -1 -1]v = 0

Solving the system of equations, we find v₂ = [1; 2; 1].

Since the matrix A has three distinct eigenvalues and we were able to find three linearly independent eigenvectors, the matrix A is diagonalizable according to Theorem 7.5.

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To graph the equation f(x) = 3x^3 + 2x, we can start by identifying the intercepts, asymptotes, max/min values, and inflection points.

Intercepts:

To find the x-intercepts, we set f(x) = 0 and solve for x:

0 = 3x^3 + 2x

By factoring out an x, we get:

0 = x(3x^2 + 2)

This equation is satisfied when x = 0 or when 3x^2 + 2 = 0. However, the quadratic equation does not have real solutions. Therefore, the only x-intercept is x = 0.

To find the y-intercept, we evaluate f(0):

f(0) = 3(0)^3 + 2(0) = 0

So the y-intercept is y = 0.

Asymptotes:

Since the degree of the numerator (3x^3 + 2x) is greater than the degree of the denominator (1), there are no horizontal asymptotes.

Max/Min Values and Inflection Points:

To find the critical points, we take the derivative of f(x):

f'(x) = 9x^2 + 2

Setting f'(x) = 0, we find the critical point:

9x^2 + 2 = 0

This equation does not have real solutions, so there are no critical points, max/min values, or inflection points.

Plotting these results on a graph, we have a straight line passing through the origin (0,0) with no intercepts, asymptotes, max/min values, or inflection points.

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The general solution for the given equation is:

y = ±√(4x² - 3)

The given equation √(y² + 3) = 2x is a quadratic equation involving the variable y. To find the general solution, we want to isolate y on one side of the equation.

To do that, we start by squaring both sides of the equation to eliminate the square root:

(y² + 3) = (2x)²

y² + 3 = 4x²

Next, we move the constant term to the other side:

y² = 4x² - 3

To solve for y, we take the square root of both sides:

y = ±√(4x² - 3)

The ± symbol indicates that we consider both the positive and negative square root solutions. So the general solution is:

y = ±√(4x² - 3)

The constant C is not included in the general solution because it represents an arbitrary constant that can take any value.

It typically arises when solving differential equations or when dealing with initial value problems, where the value of C is determined by the specific conditions or initial values given in the problem.

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The derivative of the integral of f(x) dx from 0 to any value of x is always zero.

The integral of f(x) dx, we can use the Fundamental Theorem of Calculus. According to the theorem, if F(x) is an antiderivative of f(x), then the integral of f(x) dx from a to b is given by F(b) - F(a).

In this case, we are given that the integral of f(u) du from 0 to f(x) is equal to 6. Let's express this information using the variable u:

∫[0, f(x)] f(u) du = 6

Now, we can differentiate both sides of this equation with respect to x to find the derivative of the integral:

d/dx ∫[0, f(x)] f(u) du = d/dx 6

Using the Fundamental Theorem of Calculus, the left side becomes f(f(x)) × f'(x):

f(f(x)) × f'(x) = 0

We can rewrite this equation as:

f'(x) × f(f(x)) = 0

Since we know that f(x) is continuous, we can conclude that f'(x) must be zero for all values of x in order to satisfy the equation.

Therefore, fx = f'(x) = 0 for all x.

In summary, the derivative of the integral of f(x) dx from 0 to any value of x is always zero.

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The velocity of the object when it returns to its initial position is (9 ± 3√17)/4. The object is at rest for all time values. The object moves in a positive direction for t > -1/4. The object has a constant acceleration of 12 m/s².

To find the object's acceleration at t = 2 seconds, we need to take the second derivative of the position function, s(t).

Position function: s(t) = 6t² + 3t

Taking the derivative of s(t) with respect to t:

s'(t) = 12t + 3

Taking the second derivative of s(t) with respect to t:

s''(t) = 12

Therefore, the object's acceleration at t = 2 seconds is constant and equal to 12 m/s².

To find the time(s) when the object is at rest, we need to find the value(s) of t for which the velocity is zero.

Velocity function: v(t) = s'(t) = 12t + 3

Setting v(t) = 0 and solving for t:

12t + 3 = 0

12t = -3

t = -3/12

t = -1/4

Since time cannot be negative in this context, there are no real solutions for when the object is at rest.

To determine when the object is moving in a positive direction, we need to find the intervals where the velocity function is positive (v(t) > 0).

For v(t) = 12t + 3 to be positive:

12t + 3 > 0

12t > -3

t > -3/12

t > -1/4

The object is moving in a positive direction for t > -1/4 (all positive time values).

To identify when the object is slowing down, we need to analyze the sign of the acceleration function.

Since the acceleration function, a(t), is a constant 12 m/s², it is always positive. Therefore, the object is not slowing down at any point.

To find the velocity of the object when it returns to its initial position, we need to determine when the displacement (change in position) is zero.

Displacement function: Δs(t) = s(t) - s(0) = s(t) - 6

Setting Δs(t) = 0 and solving for t:

s(t) - 6 = 0

6t² + 3t - 6 = 0

2t² + t - 2 = 0

Using the quadratic formula

t = (-b ± √(b² - 4ac)) / (2a)

a = 2, b = 1, c = -2

t = (-1 ± √(1 - 4(2)(-2))) / (2(2))

t = (-1 ± √(1 + 16)) / 4

t = (-1 ± √17) / 4

Therefore, the velocity of the object when it returns to its initial position can be found by substituting the time value into the velocity function, v(t) = 12t + 3

v((-1 ± √17) / 4) = 12((-1 ± √17) / 4) + 3

To find the exact value of the velocity when the object returns to its initial position, let's substitute the time value into the velocity function

v(t) = 12t + 3

Substituting t = (-1 ± √17) / 4:

v((-1 ± √17) / 4) = 12((-1 ± √17) / 4) + 3

Expanding and simplifying

v((-1 ± √17) / 4) = (12(-1) ± 12√17) / 4 + 3

v((-1 ± √17) / 4) = -12/4 ± 3√17/4 + 3

v((-1 ± √17) / 4) = -3 ± 3√17/4 + 12/4

v((-1 ± √17) / 4) = (-3 + 12)/4 ± 3√17/4

v((-1 ± √17) / 4) = 9/4 ± 3√17/4

Therefore, the exact velocity of the object when it returns to its initial position is given by the expression (9 ± 3√17)/4.

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The required answer is the general, or nth term, an of the sequence is 9n.

The given sequence is: 0, 9 log 10, 9 log 100, 9 log 1000, ...

We can observe that each term in the sequence is obtained by evaluating the expression 9 log ([tex]10^n[/tex]), where n is the position of the term in the sequence.

Simplifying this expression, we have:

9 log ([tex]10^n[/tex]) = 9n log 10

Since log 10 is equal to 1, we can further simplify:

9n log 10 = 9n

Therefore, the required answer is the general, or nth term, an of the sequence is 9n.

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The equation of the tangent to the graph ln x + xe = 1 at the point (3, 0) is. 3y = (1 + 3e) (x - 3).

So given the equation of the graph is,

ln x + xe = 1

So, y = ln x + xe - 1

Differentiating the above function with respect to 'x' we get,

dy/dx = d/dx [ln x + xe - 1 ] = d/dx [ln x] + e d/dx [x] - d/dx [1] = 1/x + e

So the slope of the tangent to the graph at the point (3, 0) is = dy/dx at point (3, 0) = 1/3 + e

So the equation of the tangent to the graph at the point (3, 0) is given by,

(y - 0) = (1/3 + e) (x - 3)

y = [(1 + 3e)(x - 3)]/3

3y = (1 + 3e) (x - 3)

Hence the equation of the tangent to the graph at the point (3, 0) is. 3y = (1 + 3e) (x - 3).

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The solution to the initial value problem √y dx + (x - 2)dy = 0, y(3) = 9 is given by the equation -2√y = ln|x-2| + C, where C is a constant. This equation represents the implicit solution of the problem.

To solve the given initial value problem, we can start by rearranging the equation √y dx + (x - 2)dy = 0. Separating the variables, we have √y dx = -(x - 2)dy. Next, we integrate both sides of the equation. Integrating √y with respect to x gives 2√y, and integrating -(x - 2) with respect to y gives -2(x - 2)y.

Therefore, we have 2√y = -2(x - 2)y + C, where C is the constant of integration. Simplifying further, we get -2√y = 2(x - 2)y + C. Rearranging the equation, we have -2√y = ln|x-2| + C, which represents the implicit solution of the initial value problem.

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Answer:

The range is the difference between the highest and lowest values in a dataset.

To find the range of each class, we need to find the highest and lowest values in each set.

Class 1 : Range = 4 - 2 = 2 Class 2: Range = 6 - 2 = 4T

Therefore, the relationship between the range of the two classes is that Class 2 has twice the range of Class 1.

Step-by-step explanation:

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This is the right hand derivative and the value of limit is 16 .

Given,

f(x) = x²

x = 8

Now,

Right hand derivative :

[tex]\lim_{h \to \ 0} f(a + h) - f(a)/h[/tex]

if it exists is called an right hand derivative of f(x) at x = a .

Here,

f(x) = x²

a = 8

Substitute the values,

[tex]Rf'(8) = \lim_{h \to \ 0} f(8 + h) - f(8)/h\\= \lim_{h \to \ 0} (8 + h)^2 - (8)^2/h\\= \lim_{h \to \ 0} 64 + h^2 + 16h -64/h\\\lim_{h \to \ 0} h^2 + 16h /h\\= \lim_{h \to \ 0} h + 16\\[/tex]

Now,

put the value of h

= 16 .

Hence the value of limit is 16 .

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Answer:

  about 0.167 N/cm²

Step-by-step explanation:

You want the pressure when the force resulting in a pressure of 0.25 N/cm² over an area of 200 cm² is instead applied over an area of 300 cm².

Pressure is the ratio of force to area. Then the force applied by the original box/cloth arrangement is ...

  F = (200 cm²)(0.25 N/cm²) = 50 N

When that force is applied over 300 cm², the pressure is ...

  P = F/A = (50 N)/(300 cm²) = 1/6 N/cm² ≈ 0.167 N/cm²

The reduced pressure is about 0.167 N/cm².

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The given system of linear equations can be written in matrix form as AX = B, where A is the coefficient matrix, X is the column vector of variables (x, y, z, w), and B is the column vector of constants.

a) By substituting the given values, the system can be represented as:

0x + 1y + 1z + 1w = 0

1x + 1y + 1z + 1w = -1

1x + 3y - 7z + 2w = 1

3x - 2y - 6z + 4w = 12

b) To find the solution space, we can solve the system using Gaussian elimination or row reduction. After applying the row reduction process, we can obtain the row echelon form of the augmented matrix and determine the solution space.

c) The solution space can be expressed as a vector in R, where R represents the set of real numbers. Each vector represents a specific combination of values for variables x, y, z, and w that satisfies the given system of equations. The solution space may contain a unique solution, infinitely many solutions, or no solution, depending on the outcome of the row reduction process.

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To find the derivative of the function f(x) = 3x² - 6x + 2, we can use the power rule for differentiation.

The power rule states that if we have a term of the form ax^n, the derivative with respect to x is given by nx^(n-1).

Applying the power rule to each term of f(x), we get:

f'(x) = d/dx(3x²) - d/dx(6x) + d/dx(2)

Taking the derivative of each term:

f'(x) = 2(3)x^(2-1) - 6(1)x^(1-1) + 0

Simplifying:

f'(x) = 6x - 6

Now, to find f'(1), we substitute x = 1 into the derivative:

f'(1) = 6(1) - 6 = 6 - 6 = 0

Therefore, f'(1) = 0.

The value of the expression at x = 16 is approximately -5.8942e+114.

The value of the expression at x = 2 is approximately 6.74624e+27.

The integral of sin(3r) with respect to r is -(1/3) * cos(3r) + C.

The integral of [2² * cos(2³¹)] with respect to a is [2² * sin(2³¹)] + C.

We have,

To evaluate the given expressions, let's go step by step:

1)

Evaluating the expression [tex]-2e^{x^2} x^3 + e^{1/6}[/tex]at x = 16:

Plug in the value of x into the expression:

[tex]-2e^{16^2} 16^3 + e^{1/6}[/tex]

Simplifying the exponentials:

[tex]-2e^{256} 16^3 + e^{1/6}[/tex]

Now you can calculate the exponential terms and perform the multiplication and addition/subtraction:

-2 * (approximately 7.20049e+110) * 4096 + (approximately 1.20569)

≈ -5.8942e+114 + 1.20569 ≈ -5.8942e+114.

Therefore, the value of the expression at x = 16 is approximately -5.8942e+114.

2)

Evaluating the expression [tex]16^{16x^2} / 7[/tex] at x = 2:

Plug in the value of x into the expression:

[tex]16^{16 \times 2^2 / 7[/tex]

Simplify the exponent:

[tex]16^{16 \times 4} / 7[/tex]

Calculate the exponentiation and division:

(4.72237e+28) / 7 ≈ 6.74624e+27.

Therefore, the value of the expression at x = 2 is approximately 6.74624e+27.

3)

Evaluating the integral of sin(3r) with respect to r:

The integral of sin(3r) with respect to r is given by:

-(1/3)  cos(3r) + C,

where C is the constant of integration.

4)

Evaluating the integral of [2² * cos(2³¹)] with respect to a:

The integral of [2² * cos(2³¹)] with respect to a is:

[2² * sin(2³¹)] + C,

where C is the constant of integration.

Thus,

The value of the expression at x = 16 is approximately -5.8942e+114.

The value of the expression at x = 2 is approximately 6.74624e+27.

The integral of sin(3r) with respect to r is -(1/3) * cos(3r) + C.

The integral of [2² * cos(2³¹)] with respect to a is [2² * sin(2³¹)] + C.

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The complete question:

Evaluate the expression -2e^(x²) * x³ + e^(1/6) at x = 16.

Evaluate the expression 16^(16x²) / 7 at x = 2.

Evaluate the integral of sin(3r) with respect to r.

Evaluate the integral of [2² * cos(2³¹)] with respect to a.

Answer: 15

Step-by-step explanation:

Given:

AB=BC

AB= CD

Find:   ∝

Solution:

Since AB = BC , this means that ΔABC is a 45-45-90.

<A=45

<ABC = 45

Let's use ratio of 45-45-90 to assume length ratio

AB = √2

AC =  1

BC = 1

This means for other triangle:

CD = 1

<CBD = 180 - <ABC                    > Linear pair

<CBD = 180 - 45

<CBD = 135

Use law of sin to find <D

[tex]\frac{sin D}{1} =\frac{sin < CBD}{CD} }[/tex]

[tex]sin D =\frac{sin 135}{\sqrt{2}} }[/tex]

sin <D = .5

<D = 30

∝ =  180 - <D - <CBD             > Angles of triangle =180

∝ = 180 - 30 - 135

∝ = 15

The volume of the object  is π cubic and the surface area is 2π cubic

The volume of the object obtained by rotating the curve about the x-axis.

To determine the volume of the region by this formula

[tex]V=\pi\int\limits^a_b {[f(x)}]^2} \, dx[/tex]

[tex]V = \pi\int\limits^\infty_1 {[\frac{1}{x}]^2 } \, dx[/tex]

[tex]v = \pi[\frac{1}{\infty} -\frac{1}{1} ]= \pi[/tex]

The volume of the object is π cubic unit.

To determine the surface area by this formula

[tex]s=2 \pi \int\limits^a_b {f(x)\sqrt{1+f'(x)} } \, dx[/tex]

here, f(x ) = 1/x and f'(x) = -1/x²

Plugging the value in equation

[tex]s = 2\pi \int\limits^\infty_1 {\frac{1}{x} \sqrt{1+(\frac{-1}{x^2} )} } \, dx=2\pi[/tex]

Therefore, the volume of the object  is π cubic and the surface area is 2π cubic

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Answer:

  13900

Step-by-step explanation:

You want an estimate of the total of 4352, 5821, and 3670 by rounding to hundreds.

The rounded values, in hundreds, are 44, 58, 37. The sum of these is 139 hundred, or ...

  13900

The town's population is estimated to be 13900.

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The solution to the Laplace equation V²u = 0 with the given function f(θ) as [tex]u(r, \theta) = \sum[0 * r^\lambda+ (100 * \sqrt{2}) * r^{(-\lambda)}][Ccos(\lambda\theta) + Dsin(\lambda\theta)][/tex]

To solve the Laplace equation V²u = 0 with the given boundary condition u(1, θ) = f(θ), where f(θ) is defined as:

f(θ) = [tex]\left \{ {100 {if} 0 \leq \theta < \pi/4} \atop {0 if \pi/4 < \theta < 0}} \right}} \right.[/tex]

We will use separation of variables to find the solution. Let's assume the solution can be written as u(r, θ) = R(r)h(θ), where R(r) represents the radial component and h(θ) represents the angular component.

Using separation of variables, we can write the Laplace equation as:

[tex](1/r)(d/dr)(r(dR/dr)) + (1/r^2)(d^2h/d\theta^2) = 0[/tex]

To separate the variables, we set each term equal to a constant, denoted by -λ²:

[tex](1/r)(d/dr)(r(dR/dr)) = \lambda^2 (1)[/tex]

[tex](1/r^2)(d^2h/d\theta^2) = -\lambda^2 (2)[/tex]

Solving equation (1), we obtain the radial equation:

[tex]r(d^2R/dr^2) + (dR/dr) - \lambda^2R = 0[/tex]

This is a standard differential equation with solutions of the form [tex]R(r) = Ar^{\lambda} + Br^{-\lambda}[/tex], where A and B are constants.

Solving equation (2), we obtain the angular equation:

d²Θ/dθ² + λ²Θ = 0

This is a standard differential equation with solutions of the form

Θ(θ) = Ccos(λθ) + Dsin(λθ), where C and D are constants.

Now, we can combine the radial and angular components to form the general solution:

[tex]u(r, \theta) = \sum[Ar^\lambda + Br^{-\lambda}][Ccos(\lambda\theta) + Dsin(\lambda\theta)][/tex]

Next, we apply the boundary condition u(1, θ) = f(θ):[tex]u(1, \theta) = \sum[Ar^\lambda + Br^{-\lambda}][Ccos(\lambda\theta) + Dsin(\lambda\theta)] = f(\theta)[/tex]

Comparing the terms on both sides, we can determine the coefficients A, B, C, and D using the given function f(θ).

To solve these equations, we'll use trigonometric identities and properties. Let's begin with the first equation:

[tex]Acos(\lambda\theta) + Bsin(\lambda\theta) = 100[/tex]

We can rewrite this equation using the identity sin(π/4) = cos(π/4) = [tex]\sqrt2[/tex]:

[tex]Acos(\lambda\theta) + Bsin(\lambda\theta) = 100[/tex]

(A/[tex]\sqrt2[/tex] ) * [tex]\sqrt2[/tex]  * cos(λθ) + (B/[tex]\sqrt2[/tex] ) * [tex]\sqrt2[/tex] * sin(λθ) = 100

Now, we can equate the coefficients of cos(λθ) and sin(λθ) separately to determine A and B. Since cos(λθ) and sin(λθ) are orthogonal functions, their coefficients must be zero:

(A/[tex]\sqrt2[/tex]) * [tex]\sqrt2[/tex] = 0 (coefficient of cos(λθ))

(B/[tex]\sqrt2[/tex]) * [tex]\sqrt2[/tex]= 100 (coefficient of sin(λθ))

From the first equation, we can conclude that A = 0. Substituting this into the second equation:

(B/[tex]\sqrt2[/tex]) * [tex]\sqrt2[/tex] = 100

B = [tex]\sqrt2[/tex] * 100

B = 100 * [tex]\sqrt2[/tex]

Therefore, we have A = 0 and B = 100 * [tex]\sqrt2[/tex].

For the interval π/4 < θ < 2π, we have:

[tex]Acos(\lambda\theta) + Bsin(\lambda\theta) = 0[/tex]

Again, equating the coefficients of cos(λθ) and sin(λθ) separately:

[tex]Acos(\lambda\theta) + Bsin(\lambda\theta) = 0[/tex]

A * 0 + B * 1 = 0

B = 0

From this equation, we conclude that B = 0.

To summarize, we found A = 0, B = 100 * [tex]\sqrt2[/tex] for the interval 0 ≤ θ < π/4, and B = 0 for the interval π/4 < θ < 2π.

Using these coefficients, we can now write the solution to the Laplace equation V²u = 0 with the given function f(θ) as:

[tex]u(r, \theta) = \sum[0 * r^\lambda+ (100 * \sqrt{2}) * r^{(-\lambda)}][Ccos(\lambda\theta) + Dsin(\lambda\theta)][/tex]

Therefore, the solution to the Laplace equation V²u = 0 with the given function f(θ) as [tex]u(r, \theta) = \sum[0 * r^\lambda+ (100 * \sqrt{2}) * r^{(-\lambda)}][Ccos(\lambda\theta) + Dsin(\lambda\theta)][/tex]

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The Fourier transform of the function f(x) = 2x + 5, defined as F(f)(ξ) = ∫[infinity] e^(-2πixξ) f(x) dx, needs to be calculated.

To find the Fourier transform of the given function f(x) = 2x + 5, we need to evaluate the integral F(f)(ξ) = ∫[infinity] e^(-2πixξ) f(x) dx.

However, it seems there is a discrepancy in the provided information. The function f(x) is defined as 2x + 5 for the interval 2 ≤ x ≤ 20, and "otherwise" it is not specified. Without further information about the behavior of the function outside the interval, it is not possible to determine the Fourier transform accurately.

The Fourier transform involves integrating the function over its entire domain, and if the behavior of the function is not known outside the given interval, it affects the calculation of the integral and thus the Fourier transform.

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(x,y) = x² + y² -5x-9y-xy+90239 for all x and y. ≥39²3/, The inequality f(x, y) ≥ 39²3/ holds for all x and y.

To prove that f(x, y) ≥ 39²3/ for all x and y, we need to show that the given function f(x, y) = x² + y² - 5x - 9y - xy + 90239 is greater than or equal to 39²3/ for all values of x and y.

Let's simplify the expression by collecting like terms:

f(x, y) = x² - xy - 5x + y² - 9y + 90239

Now, to show that f(x, y) ≥ 39²3/, we can complete the square for the x and y terms: f(x, y) = (x² - 2.5x + 1.25²) - 1.25² - xy - 9y + y² + 90239

Expanding and rearranging, we have:

f(x, y) = (x - 1.25)² + (y - 4.5)² + y² - xy - 1.25² + 90239

Notice that the expression (x - 1.25)² + (y - 4.5)² is always non-negative since it represents the sum of two squares. Therefore, the entire expression f(x, y) will be greater than or equal to the constant term - 1.25² + 90239, which is equal to 39²3/. Hence, f(x, y) ≥ 39²3/ for all x and y.

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The solution of the given initial value problem. ty' + (t+1) y = t, y(ln 8) = 1, t> 0 is [tex]\\y = 1 - 1/t + (8/(ln 8)) * e^(^-^t^)[/tex]

To solve the given initial value problem, we'll use the method of integrating factors.

The given differential equation is:

ty' + (t + 1)y = t

Rewrite the equation in the standard form:

y' + (t + 1)/t * y = 1

The integrating factor (IF) is given by the exponential of the integral of the coefficient of y, which in this case is (t + 1)/t:

[tex]IF = te^t + Ce^t[/tex]

Multiplying both sides of the differential equation by the integrating factor, we have:

[tex]te^t * y' + (t + 1)e^t * y = te^t[/tex]

Now, we can simplify the left side using the product rule:

[tex](te^t * y)' = te^t[/tex]

After integrating both sides with respect to t, we obtain:

[tex]= te^t - e^t + C[/tex]

Dividing both sides by [tex]te^t[/tex], we get:

[tex]y = 1 - 1/t + Ce^(^-^t^)[/tex]

Now, we can use the initial condition y(ln 8) = 1 to find the value of C.

Substituting ln 8 for t and 1 for y in the equation, we have:

[tex]1 = 1 - 1/(ln 8) + Ce^(^-^(^l^n^ 8^)^)\\1 = 1 - 1/(ln 8) + C/8[/tex]

Simplifying:

[tex]0 = -1/(ln 8) + C/8[/tex]

Multiplying through by 8(ln 8), we get:

[tex]0 = -8 + C(ln 8)\\C(ln 8) = 8\\C = 8/(ln 8)[/tex]

Therefore, the solution to the initial value problem is:

[tex]y = 1 - 1/t + (8/(ln 8)) * e^(^-^t^)[/tex]

The solution is valid for t > 0.

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The number of ways to form a committee with 25 delegates, taking one person from each country, can be calculated as the product of the number of choices from each country.

Since there are 5 delegates from each of the 100 countries, the total number of ways is (5^100), which can be expressed as an extremely large number.

1. Each country has 5 delegates, and we need to select 1 delegate from each country. So, we have to make 100 choices.

2. For each choice, we have 5 options (as there are 5 delegates from each country).

3. By multiplying the number of choices for each country, we get the total number of ways: 5^100.

4. Calculating 5^100 gives us a very large number, which is the total number of ways to form the committee with 25 delegates.

Note: The actual numerical value of 5^100 is too large to be expressed accurately, but it can be approximated using scientific notation or calculated using specialized tools.

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We need to verify the equality 1 [f(x) + g(x)] dx = f(x) dx + g(x) dx for each pair of functions given: (a) f(x) = x, g(x) = 1, (b) f(x) = 1, g(x) = x², (c) f(x) = e^x, g(x) = 1, and (d) f(x) = x².

(a) f(x) = x, g(x) = 1:

Using the properties of integration, we have ∫[f(x) + g(x)] dx = ∫(x + 1) dx = (x²/2 + x) + C.

Also, ∫f(x) dx = ∫x dx = x²/2 + C₁, and ∫g(x) dx = ∫1 dx = x + C₂.

Adding the integrals, we get f(x) dx + g(x) dx = (x²/2 + C₁) + (x + C₂) = x²/2 + x + C.

Therefore, 1 [f(x) + g(x)] dx = f(x) dx + g(x) dx is verified.

(b) f(x) = 1, g(x) = x²:

Using the same approach, we have ∫[f(x) + g(x)] dx = ∫(1 + x²) dx = (x + x³/3) + C.

Also, ∫f(x) dx = ∫1 dx = x + C₁, and ∫g(x) dx = ∫x² dx = x³/3 + C₂.

Adding the integrals, we get f(x) dx + g(x) dx = (x + C₁) + (x³/3 + C₂) = x + x³/3 + C.

Therefore, 1 [f(x) + g(x)] dx = f(x) dx + g(x) dx is verified.

(c) f(x) = e^x, g(x) = 1:

Using the same approach, we have ∫[f(x) + g(x)] dx = ∫(e^x + 1) dx = (e^x + x) + C.

Also, ∫f(x) dx = ∫e^x dx = e^x + C₁, and ∫g(x) dx = ∫1 dx = x + C₂.

Adding the integrals, we get f(x) dx + g(x) dx = (e^x + C₁) + (x + C₂) = e^x + x + C.

Therefore, 1 [f(x) + g(x)] dx = f(x) dx + g(x) dx is verified.

(d) f(x) = x²:

Using the same approach, we have ∫[f(x) + g(x)] dx = ∫(x² + x²) dx = (2x³/3) + C.

Also, ∫f(x) dx = ∫x² dx = x³/3 + C₁, and ∫g(x) dx = ∫x² dx = x³/3 + C₂.

Adding the integrals, we get f(x) dx + g(x) dx = (x³/3 + C₁) + (x³/3 + C₂) = 2x³/3 + C.

Therefore, 1 [f(x) + g(x)] dx = f(x) dx + g(x) dx is verified.

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