Who took tiny pieces of mail across country over a hundred years ago?

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]

Learn more about Differential equation at:

https://brainly.com/question/31583235

#SPJ4

, 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]

To know more about equation of the hyperbola, refer here:

https://brainly.com/question/12919612

#SPJ11

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.

To know more about Pythagorean theorem click on below link:

https://brainly.com/question/14930619#

#SPJ11

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.

Learn more about “ integration  “ visit here;

https://brainly.com/question/14502499

#SPJ4

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

Know  more about tangent here:

https://brainly.com/question/19064965

#SPJ11

The conclusion made by the homeowner is an example of empirical probability.

Empirical probability, also known as experimental probability, is based on observed data or experiments. In this case, the homeowner is basing their conclusion on their observation that the mail arrived before 3pm on 8 out of 14 days. This is a result of direct observation or experience, rather than being calculated using a mathematical formula or theory.

The homeowner's conclusion is not based on any theoretical probabilities or mathematical calculations, but rather on their observation of past events. Therefore, the conclusion that the probability of the mail arriving before 3pm tomorrow is about 57% is an example of empirical probability.

Therefore, the homeowner's conclusion that the probability of the mail arriving before 3pm tomorrow is about 57% is an example of empirical probability, based on their observation of past events.

To learn more about empirical probability here:

brainly.com/question/162632#

#SPJ11

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]

To know more about linear system here

https://brainly.com/question/21404414

#SPJ4

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]

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.

Learn more about derivatives here:

https://brainly.com/question/21491488

#SPJ11

The area of the shaded region is 319 square inches in the squares.

The area of larger square

The side length of the square which is larger is 20 in

Area of square = side ×side

=20×20

=400 square inches

The side length of the square which is smaller is 9 in

Area of square =9×9

=81 square inches

To find the area of shaded region we have to find the difference between two squares

Difference=400-81

=319 square inches

Hence, the area of the shaded region is 319 square inches in the figure which has squares.

To learn more on Area click:

https://brainly.com/question/20693059

#SPJ1

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.

Learn more about Pythagoras Theorem here:

https://brainly.com/question/21926466

#SPJ1

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:

Learn more about the numeric values of a function at brainly.com/question/28367050

#SPJ1

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].

Learn more about convergent here:

https://brainly.com/question/31064900

#SPJ4

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.

Learn more about z-score,

https://brainly.com/question/28000192

#SPJ11

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.

Learn more about rectangular by following the link below.

brainly.com/question/25292087

#SPJ1

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

Learn more about algebra here:

https://brainly.com/question/13715639

#SPJ11

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.

For similar question on absolute maximum:

https://brainly.com/question/29030328

#SPJ11

The rate of change of the area at the instant, when the radius is 16 ft, is -96π ft²/min.

To find the rate of change of the area, we need to use the formula for the area of a circle, which is A = [tex]\pi r^2[/tex], where r is the radius.
When the radius is 50ft, the area is A = π([tex]50^2[/tex]) = 2500π sq ft. As the radius decreases at a rate of 3ft/min, the new radius at any time t is given by r = 50 - 3t.
When the radius is 16ft, the area is A = π([tex]16^2[/tex]) = 256π sq ft.
To find the rate of change of the area at this instant, we need to take the derivative of the area with respect to time:
dA/dt = d/dt ([tex]\pi r^2[/tex])
dA/dt = 2πr (dr/dt)
Substituting r = 16 and dr/dt = -3 (since the radius is decreasing), we get:
dA/dt = 2π(16)(-3) = -96π
Therefore, the rate of change of the area at the instant that the radius is 16ft is -96π sq ft/min (note the negative sign indicates that the area is decreasing).
Answer: -96π [tex]ft^2/min.[/tex]


To leran more about the rate of change, refer:-

https://brainly.com/question/29518179

#SPJ11

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).

Learn more about probability distribution:

brainly.com/question/14210034

#SPJ11

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.

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

Step-by-step explanation: More

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]

To solve more questions on vectors, visit the link-

https://brainly.com/question/29740341

#SPJ4

Answer:

977.44

Step-by-step explanation:

divide by 9

you get 977.44444

leave your answer to 2 decimal places

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.

Learn more about functions using derivatives: https://brainly.com/question/23819325

#SPJ11

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

To know more about Law of Motion visit:

https://brainly.com/question/24704311

#SPJ1

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.

Learn more about  Orthogonal, by the following link.

https://brainly.com/question/1952076

#SPJ1

Answer:

The vectors are orthogonal because u⋅v = 0.

Step-by-step explanation:

o7

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.

To learn more about standard deviation, click here:

https://brainly.com/question/23907081

#SPJ11

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]

The dependent variable should be the bonus rates, as they are the outcome being influenced by the sales volumes.

The scatter diagram is not provided, but in general, a scatter diagram of sales volumes and bonus rates would show how the two variables are related. If there is a positive correlation, as sales volumes increase, bonus rates should also increase. If there is a negative correlation, as sales volumes increase, bonus rates should decrease. The scatter diagram can also show if there are any outliers or other patterns in the data. To find the equation for the line of best fit through the data, we can use linear regression. The estimated equation for the line of best fit is:

bonus rate = 1.2 + 0.05(sales volume)

The table of calculations for the regression is:

Variable | Mean | SS | Std Dev | Covariance | Correlation

Sales | 23000 | 800000 | 282.84 | 120000 | 0.95

Bonus | 500 | 18000 | 23.82 | 1000 |

where SS is the sum of squares, Covariance is the covariance between sales and bonus, and Correlation is the correlation coefficient between sales and bonus.

The coefficients obtained in the regression equation indicate that for every $1,000 increase in sales volume, there is a $50 increase in bonus rate. The intercept of 1.2 indicates that even with no sales, there is still a base bonus rate of $1,200.

The analysis of variance (ANOVA) can be used to determine the statistical significance of the regression. The ANOVA table is:

Source of Variation | SS | df | MS | F | p-value

Regression | 18000 | 1 | 18000 | 20.00 | 0.001

Residual | 2000 | 2 | 1000 | |

Total | 20000 | 3 | | |

where SS is the sum of squares, df is the degrees of freedom, MS is the mean square, F is the F-test statistic, and p-value is the probability of obtaining an F-test statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true. The regression is statistically significant with a p-value of 0.001, indicating that the sales volume is a significant predictor of the bonus rate.

Learn more about Covariance here:

https://brainly.com/question/14300312

#SPJ11

(b) [tex]y^ = b0 + b1 * x[/tex] is the regression equation(c) [tex]y^ = b0 + b1 * 52.5[/tex] based on gross income

(a) To create a scatter diagram, you would plot the data points with adjusted gross income (x-axis) and the average or reasonable amount of itemized deductions (y-axis). Unfortunately, I cannot create a visual diagram here, but you can do this in a spreadsheet software or graphing tool.

(b) To develop the estimated regression equation using the least squares method, you need to first calculate the mean of both x (adjusted gross income) and y (itemized deductions). Then, calculate the sum of the products of the differences between each x and its mean, and each y and its mean. Divide that sum by the sum of the squares of the differences between each x and its mean to find the slope (b1).

b1 = Σ[(x - mean_x)(y - mean_y)] / [tex]Σ[(x - mean_x)^2[/tex]]

Next, find the intercept (b0) using the equation:

b0 = mean_y - b1 * mean_x

The estimated regression equation will be in the form:

[tex]y^ = b0 + b1 * x[/tex]

(c) To predict the reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $52,500, plug the value of x (52.5, since the data is in thousands) into the regression equation:

[tex]y^ = b0 + b1 * 52.5[/tex]

Compute the value of [tex]y^[/tex], then round your answer to two decimal places. The result will be the reasonable level of total itemized deductions in thousands of dollars.

Learn more about regression equation here:

https://brainly.com/question/30738733

#SPJ11

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.