I'll give brainliest

Answer:

11° 33'

Step-by-step explanation:

[tex]21' = \left(\dfrac{21}{60}\right)^\circ = \left(\dfrac{7}{20}\right)^\circ \\50^\circ 21' = 50 \dfrac{21}{60}\; degrees\\\\38^\circ 48' = 38 \dfrac{48}{60} \;degrees[/tex]

Break
[tex]50 \dfrac{21}{60} = 49 + 1 \dfrac{21}{60}\\[/tex]

[tex]1 \dfrac{21}{60} = \dfrac{1 \times 60 + 21}{60} = \dfrac{81}{60}[/tex]

Therefore
[tex]50^\circ21' - 38^\circ48' = 49\dfrac{81}{60} - 38\dfrac{48}{60}[/tex]

Subtract the whole numbers:
49 - 38 = 11

Subtract the fractions:
[tex]\dfrac{81}{60} - \dfrac{48}{60} = \dfrac{33}{60} = 33'[/tex]

So

50°21' - 38°48' = 11° 33'

[tex]\textit{Partial Variation} \\\\ y = k_ox+k_1\hspace{5em}\textit{"y" varies partly with a constant and directly with "x"} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{"D" varies partly with a constant and directly with "V"}}{D=k_oV+k_1}\qquad \textit{we know that} \begin{cases} V=40\\ D=150\\[-0.5em] \hrulefill\\ V=54\\ D=192 \end{cases}[/tex]

[tex]\boxed{\begin{array}{llll} 150=40k_o+k_1\\\\ 192=54k_o+k_1 \end{array}}\qquad \stackrel{ \textit{using elimination method} }{\begin{array}{llll} -150=-40k_o-k_1\\\\ ~~ 192= ~~ 54k_o+k_1\\\cline{1-1} ~~ 42~= ~~ 14k_o+0 \end{array} }\qquad \implies 42=14k_o \\\\\\ \cfrac{42}{14}=k_o\implies \boxed{3=k_o}\hspace{5em}\stackrel{\textit{substituting on the 1st equation}}{150=40(3)+k_1}[/tex]

[tex]\cfrac{150}{40(3)}=k_1\implies \boxed{\cfrac{5}{4}=k_1}\hspace{5em} {\Large \begin{array}{llll} D=3V+\cfrac{5}{4} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \textit{when V=73, what is "D"?}\qquad D=3(73)+\cfrac{5}{4}\implies D=\cfrac{881}{4}\implies D=220\frac{1}{4}[/tex]

Answer:

x = 40

Step-by-step explanation:

Alternate exterior angles are congruent (equal).

2x + 16 = 96   Subtract 16 from both sides

2x + 16 -16 = 96 - 16

2x = 80  Divide both sides by 2

x = 40

Answer:

x=40

Step-by-step explanation:

The resulting values of the given operations are listed below.

Arithmetic operations are the basic part of mathematics which combines the basic operations like addition, subtraction, multiplication and division.

Given are various operations.

(1) 235,654 ÷ 10000

When dividing a number with any powers of 10, we will shift the decimal point from right to left up to the number of zeroes in the given power of 10.

Here 10,000 contains 4 zeroes.

235,654 can be written as 235,654.00.

Shift the decimal point 4 places from left to right.

235,654 ÷ 10000 = 23.5654 ≈ 23.57 rounded to two decimal places.

(2) 895,21 x 100

When multiplying a number with any powers of 10, we will shift the decimal point from left to right up to the number of zeroes in the given power of 10.

895,21 x 100 = 89521.00 x 100 = 895,2100

(3) 1307 in words is one thousand three hundred and seven.

(4) 1307 in expanded form.

1307 = (1 × 1000) + (3 × 100) + (0 × 10) + (7 × 1)

        = 1000 + 300 + 0 + 7

(5) 1307 + 13 = 1320

(6) In roman numerals, 1000 is M, 100 is C and 10 is X.

1320 = (1 × 1000) + (3 × 100) + (2 × 10) + (0 × 1)

        = M CCC XX

1320 in roman numerals is MCCCXX.

Hence all the operations are found.

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We have shown that the product of two rational numbers is rational.

A number system is defined as a system of writing to express numbers.

Let a and b be two rational numbers, such that:

a = p/q

b = r/s

where p, q, r, and s are integers and q, s are non-zero.

We need to show that the product ab is also a rational number.

We have:

ab = (p/q) x (r/s)

= (p x r) / (q x s)

Since p, q, r, and s are all integers,

ab = m/n

where m = p x r and n = q x s.

Since both m and n are integers and n is non-zero ,we can conclude that ab is a rational number.

Therefore, we have shown that the product of two rational numbers is rational.

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Answer:

3

Step-by-step explanation:

9-7=2

2*3=6

6*2=12

75/5=15

15-12=3

Answer:

Therefore, the value of the expression 75÷5−[(9−7)×3]×2 is 138.

Step-by-step explanation:

Using the order of operations (also known as PEMDAS), we simplify the expression as follows:

First, we solve the parentheses: (9-7) = 2.

Next, we multiply 2 by 3: 2 x 3 = 6.

Then, we subtract 6 from 75: 75 - 6 = 69.

Finally, we multiply 69 by 2: 69 x 2 = 138.

Therefore, the value of the expression 75÷5−[(9−7)×3]×2 is 138.

M'(11, -8), L'(13, -8), K'(13, -10) and J'(11, -10) are the coordinates of the vertices after a reflection over the line x = 4

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

The coordinates of the quadrilateral MLJK are M(7, -8), L(9, -8), K(9, -10)

and J(7,-10)

Now we have to find the  coordinates of the vertices after a reflection over the line x = 4

We need to find the coordinates of the vertices after reflecting over the line x = 4, we just need to flip the x-coordinate of each vertex across the line.

M(7, -8)--->M'(7+4, -8)

=M'(11, -8)

L(9, -8)--->L'(9+4, -8)

=L'(13, -8)

K(9, -10)--->K'(9+4, -10)

=K'(13, -10)

J(7,-10)--->J'(7+4, -10)

=J'(11, -10)

Hence, M'(11, -8), L'(13, -8), K'(13, -10) and J'(11, -10) are the coordinates of the vertices after a reflection over the line x = 4

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the probability of selecting a card that is not prime from the set {4, 5, 6, 7, 8} is 0.6 or 60%.

How we solved?

To determine the probability of selecting a card that is not prime from the set {4, 5, 6, 7, 8}, we first need to identify which numbers in the set are prime.

The prime numbers in the set are 5 and 7, so the non-prime numbers are 4, 6, and 8.

The total number of cards in the set is 5.

Therefore, the probability of selecting a non-prime card is:

P(not prime) = number of non-prime cards / total number of cards

P(not prime) = 3 / 5

P(not prime) = 0.6 or 60%

So the probability of selecting a card that is not prime from the set {4, 5, 6, 7, 8} is 0.6 or 60%.

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Answer:

the efficiency of the generator is 81.82%.

Step-by-step explanation:

The efficiency of a generator is defined as the ratio of the electrical power output to the mechanical power input. In this problem, the electrical power output is given as 90 hp, and the mechanical power input is the power supplied by the engine, which is 110 hp. Therefore, the efficiency of the generator can be calculated as follows:

Efficiency = Electrical power output / Mechanical power input

Efficiency = 90 hp / 110 hp

Simplifying the fraction, we get:

Efficiency = 0.8182

To convert this decimal to a percentage, we multiply by 100:

Efficiency = 0.8182 * 100

Rounding to two decimal places, we get:

Efficiency = 81.82%

Therefore, the efficiency of the generator is 81.82%.

The difference quotient expression for the given function is

[tex]\frac{f(x+h)-f(x)}{h} =\frac{\sqrt{(x+h+1)(x+h-1)}-\sqrt{(x+1)(x-1)} }{h}[/tex]

The expression in single-variable calculus is usually referred to as the difference quotient.

[tex]\frac{f(x+h)-f(x)}{h}[/tex]

When taken to the limit as h gets closer to zero, h frac f(x+h)-f(x)h, which gives the derivative of the function f.

The slope of a secant line passing through the curve of f(x) is measured by the difference quotient.

Consider the difference quotient formula,

[tex]\frac{f(x+h)-f(x)}{h}[/tex]

Evaluate the function at x = x + h

replace the variable x with (x + h) in the given expression

[tex]f(x+h)=\sqrt{(x+h)^2-1}[/tex]

simplify the result ,

[tex]f(x+h)=\sqrt{(x+h+1)(x+h-1)}[/tex]

find the components of the definition,

[tex]f(x+h)=\sqrt{(x+h+1)(x+h-1)}[/tex]

[tex]f(x)=\sqrt{(x+1)(x-1)}[/tex]

plug in the components,

[tex]\frac{f(x+h)-f(x)}{h} =\frac{\sqrt{(x+h+1)(x+h-1)}-\sqrt{(x+1)(x-1)} }{h}[/tex]

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The mean age of the 12 students is approximately 17.02 years, the variance is 37.41, and the standard deviation is approximately 6.117 years.

The statistic is the study of mathematics that deals with relations between comprehensive data.

Here,
To calculate the mean, variance, and standard deviation of the ages of the 12 students, we first need to find the sum of the ages:

Sum = 16 + 17 + 18 + (16/0.5) + 17 + 18 + 19 + 17 + 17 + 13 + (17/2/2) + 16

= 16 + 17 + 18 + 32 + 17 + 18 + 19 + 17 + 17 + 13 + 4.25 + 16

= 204.25

Mean = Sum / Number of students = 204.25 / 12 ≈ 17.02 years

To calculate the variance, we need to subtract the mean from each age, square the differences, add up the squares, and divide by the number of students minus one:

Variance = [ (16 - 17.02)² + (17 - 17.02)² + ... + (16 - 17.02)² ] / (12 - 1)

= [ 0.16 + 1.96 + ... + 0.16 ] / 11

= 37.41

Finally, we can calculate the standard deviation by taking the square root of the variance:

Standard deviation = √(Variance) ≈ 6.117

Therefore, the mean age of the 12 students is approximately 17.02 years, the variance is 37.41, and the standard deviation is approximately 6.117 years.

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Given the Figure shown below is, ΔABC~ΔAXY if so, 1.67 is the scale factor between the two triangles.

The ratio between the scale of an original thing and a new object that is a representation of it but has a different size is known as a scale factor (bigger or smaller).

Consider two comparable squares as an illustration. One has a side length of 2 inches, while the other has a side length of 4 inches. This results in a scale factor of 1: 2 from the small square to the huge square. The scale factor between these two comparable squares is 1: 2, or tiny square to huge square.

Given that,

12 / 20 = 17 / x

or, 12 x = 340

or, x = 28.33

on the other hand,

25 / 15 = 1.66

20 / 12 = 1.66

28.33 / 17 = 1.66

now, scale factor = 1.67

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Answer A: Construct a two-sample z-interval for the difference between population proportions.

What is fraction?

A fraction is a mathematical expression that represents a part of a whole or a ratio between two quantities. It is written as a/b, where a is called the numerator and b is called the denominator. The numerator represents the number of parts or units being considered, while the denominator represents the total number of parts or units in the whole.

According to given conditions:

The best method for the researcher to use to estimate the true difference between the population proportions would be to:

Answer A: Construct a two-sample z-interval for the difference between population proportions.

This is the most appropriate method for estimating the difference between population proportions, as it takes into account the variability of the sample proportions and provides a confidence interval for the true difference between the two populations. The formula for calculating the two-sample z-interval for the difference between population proportions is:

(p₁−p₂) ± z x √[(p₁(1−p₁))/₁ + (p₂(1−p₂))/₂]

Therefore, Answer A: Construct a two-sample z-interval for the difference between population proportions.

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The set of data that is skewed-right, there is at least 99.7% of the data within 3 standard deviations.

A statistical principle known as the empirical rule, also known as the three-sigma rule or 68-95-99.7 rule, holds that with a normal distribution, almost all observed data will lie within three standard deviations (denoted by ) of the mean or average (denoted by ).

Some points about standard deviation using Empirical Formula:

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Answer:
The perimeter of the picture frame is 44 inches.

Step-by-step explanation:
The formula for the perimeter of a rectangle is:

Perimeter = 2 x (length + width)

In this case, the length of the picture frame is 14 inches and the width is 8 inches. Substituting these values into the formula, we get:

Perimeter = 2 x (14 + 8) = 2 x 22 = 44 inches

Therefore, the perimeter of the picture frame is 44 inches.


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We expect that, on average, 1 out of 25 patients calling in claiming to have the flu will actually have the flu.

According to the nurse's observation, there is a 4% chance that a patient who calls the medical advice line and claims to have the flu truly has.

As a result, we anticipate that 4 out of every 25 people who phone in will truly have the flu.

To figure this out, we can use the formula below:

Probability of having the flu multiplied by the total number of patients equals the anticipated number of flu patients.

Estimated number of influenza patients = 0.04 times 25

Expected number of influenza patients is one.

Hence, on average, we anticipate that 1 out of every 25 individuals who call in and claim to have the flu will truly have it.

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[-1.5, 3.5] and [-2, infty) are the range and domain of the equation.

Quadratic function are functions that have a leading degree of 2.

The domain of the function are values that lies along the x-axis of the graph. Hence the domain of the curve will be [-1.5, 3.5]

The range of the function are values that lies along the y-axis of the graph. Hence the domain of the curve will be [-2, infty)

Hence the range and domain of the graph are [-1.5, 3.5] and [-2, infty)

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The value of the transformed function after a vertical stretch in the y direction is h ( x ) = k ( | x | - 3 ) , where k is the scale factor

The transformation of a function may involve any change.

Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs), etc.

If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:

Horizontal shift (also called phase shift):

Left shift by c units: y=f(x+c) (same output, but c units earlier)

Right shift by c units: y=f(x-c)(same output, but c units late)

Vertical shift:

Up by d units: y = f(x) + d

Down by d units: y = f(x) - d

Stretching:

Vertical stretch by a factor k: y = k × f(x)

Horizontal stretch by a factor k: y = f(x/k)

Given data ,

Let the parent function be represented as f ( x )

Now , the value of f ( x ) is

f ( x ) = ( | x | - 3 )   be equation (1)

Now , when the function is stretched in the y direction , we get

Vertical stretch by a factor k: y = k × f(x)

On simplifying , we get

Let the transformed function be represented as h ( x )

where the value of h ( x ) is

Let the scale factor of stretching be k

h ( x ) = k ( | x | - 3 )

Hence , the transformed function is h ( x ) = k ( | x | - 3 )

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Answer:

Therefore, the total value of Mary's investment after two years is $700.

Step-by-step explanation:

To calculate the total value of Mary's investment after two years, we need to calculate the value of the investment after the first year, and then use that value to calculate the value after the second year.

In the first year, the value of the investment increases by 8%, so the value after the first year is:

Value after first year = $600 + 8% of $600

Value after first year = $600 + 0.08 * $600

Value after first year = $648

In the second year, the value of the investment increases by another 8%, so the value after the second year is:

Value after second year = Value after first year + 8% of Value after first year

Value after second year = $648 + 0.08 * $648

Value after second year = $699.84

Rounding to the nearest dollar, the total value of Mary's investment after two years is:

Total value after two years = $700

Answer:

The ordered pair that is a solution is (2, 10)

Step-by-step explanation:

2x + 6 = 4x + 2  Subtract 2x from both sides

2x - 2x + 6 = 4x - 2x + 2

6 = 2x + 2  Subtract 2 from both sides

6 - 2 = 2x + 2 -2

4 = 2x  Divide both sides by 2

2 = x

Substitute 2 for x in either of the two original equations.

y = 2x + 6

y = 2(2) + 6

y = 4 + 6

y = 10

The ordered pair that is a solution is (2, 10)

The kind of evidence geologists use to make geological timelines of events in Earth's history is the fossil remains.

A geological timeline shows a calendar of events in Earth's history.

To develop geological timelines, scientists understudy physical objects or structures on the Earth.  These objects or structures enable them to prove a phenomenon of interest.

Generally, the best form of evidence for the study of the history of life on earth is fossils, as they give clues about geological events and past climates, including the ages of rocks.

Other kinds of evidence for developing geological timelines include the following:

Thus, fossils provide the best kind of evidence for geological timelines.

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A variable in mathematics is a number that can change depending on the issue. Several mathematical statements and equations use the generic letters x, y, and z. In other words, a variable is a symbol that stands for an unknowable numeric value. Say that x Plus 5 equals 10. The "x" variable is used here.

The variable "model of car driven" is qualitative because it is an attribute characteristic. Qualitative variables are those that are not numerical in nature but rather represent characteristics or categories that can be observed or measured through attributes. In this case, the model of the car driven is a type of attribute that describes the make and model of the car, such as Honda Civic, Toyota Corolla, etc.

Therefore, option B is the correct answer.

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Optiοn B, "The variable is qualitative because it is an attribute characteristic," is the cοrrect answer.

A qualitative variable is a type οf variable in statistics that describes a characteristic οr quality οf an οbject οr phenοmenοn. It is alsο called a categοrical variable, and it can be nοminal οr οrdinal. Nοminal variables have categοries that have nο inherent οrder, while οrdinal variables have categοries with a natural οrder οr ranking.

The variable "Mοdel οf car driven" is qualitative because it is an attribute characteristic. Qualitative variables are variables that are nοt numerical in nature but are based οn attributes οr characteristics. In this case, the mοdel οf the car driven is a categοrical variable that describes a particular characteristic οf the car, such as its make οr mοdel. This variable cannοt be measured quantitatively οr numerically, but rather it can be categοrized based οn specific attributes οr features.

Therefοre, οptiοn B, "The variable is qualitative because it is an attribute characteristic," is the cοrrect answer.

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Answer:

No solution

Step-by-step explanation:

2x - 4y= -6 ----- eq(1)

-x + 2y = -2 ----- eq(2)

Simplifying eq(1) by taking 2 as common,

2(x-2y) = 2(-3)

x - 2y= -3 -----eq(1)

from eq(1)

x - 2y = -3

x = -3 + 2y ---eq(3)

substituting eq(3) into eq(2)

-(-3 + 2y) + 2y = -2

3 - 2y + 2y = -2

3 = -2

Therefore This pair of linear Equations Has No Solution

The standard error is of 0.2246, which means that the mean scores for samples of 30 vary around 0.2246 from the mean.

The confidence interval is the range of values that you expect your estimate to fall between a certain percentage of the time if you run your experiment again or re-sample the population in the same way.

The confidence interval formula is [tex]\bar x\pm t\frac{s}{\sqrt{n}}[/tex].

Where, [tex]\bar x[/tex] is the sample mean, t is the critical value, n is the sample size and s is the standard deviation for the sample.

Here,

[tex]\bar x[/tex] =1.267, s=1.23, n=30

The standard error is Se= 1.23/√30

= 0.2246

Item a:

The standard error is of 0.2246, which means that the mean scores for samples of 30 vary around 0.2246 from the mean.

Item b:

Using a t-distribution calculator, considering a confidence level of 0.99 with 30 - 1 = 29 df, the critical value is t = 2.7564.

[tex]\bar x\pm tS_c[/tex]

[tex]\bar x+ tS_c[/tex] =1.267-2.7564(0.2246) =0.6479

[tex]\bar x- tS_c[/tex] =1.267+2.7564(0.2246) =1.8861

Therefore, the 99% confidence interval to estimate the true mean score on this question is (0.6479, 1.8861). It means that we are 99% that the true mean score of all students in this question is between 0.6479 and 1.8861.

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We kindly invite to see the image attached below to know the representation of the parabola according to its features.

A parabola is the graphic representation of any quadratic equation, that is, a polynomial of the form y = a · x² + b · x + c, where a, b, c are real coefficients. An alternative presentation is the vertex form:

y - k = C · (x - h)²

Where:

The procedure to sketch the parabola according to its features is described below:

Step 1 - Determine the y-intercept:

y = 6 · 0² - 17 · 0 + 12

y = 12

Step 2 - Determine the x-intercepts.

6 · x² - 17 · x + 12 = 0

6 · [x² - (17 / 6) · x + 2] = 0

6 · (x - 3 / 2) · (x - 4 / 3) = 0

x = 3 / 2 or x = 4 / 3

Step 3 - Determine the coordinates of the vertex:

y = 6 · x² - 17 · x + 12

y = 6 · [x² - (17 / 6) · x + 2]

y + 6 · (1 / 144) = 6 · [x² - (17 / 6) · x + 2 + 1 / 144]

y + 1 / 24 = 6 · [x² - (17 / 6) · x + 289 / 144]

y + 1 / 24 = 6 · (x - 17 / 12)²

(h, k) = (17 / 12, - 1 / 24)

Steps 4 & 5 - Add the points on Cartesian plane and match the resulting curve.

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Answer:

Step-by-step explanation:

(3x + 7y)(3x - 7y)  FOIL

9x² -49y²   yes, this is difference of two squares

(2x - 7y)(2x - 7y)   FOIL

4x² -14x -14x +49y²

4x² -28x + 49y²   not difference of two squares

9x²(5x - 7)    Distribute the 9x²

45x³ - 63x²   not difference of two squares because one term is cubed

(6x - 4y)²    Distribute the square term

36x² - 16y²   yes, difference of two squares

Answer:

This scenario describes a business combination in which Alfonso Inc. acquired 100% ownership of BelAire Company on January 1, 2020, using both cash and common stock as consideration. The combination is a statutory merger, meaning that BelAire is dissolved as a legal corporation and its assets and liabilities are assigned to a new reporting unit.

To record this business combination, Alfonso would need to perform a detailed analysis of the fair values of the assets and liabilities of BelAire as of the acquisition date, and allocate the total consideration paid (i.e., $387,500 in cash plus the fair value of 100,000 shares of Alfonso's common stock) to those assets and liabilities based on their fair values.

Assuming that the fair values of the assets and liabilities of BelAire on the acquisition date were as follows:

Assets:

Cash: $50,000

Accounts receivable: $100,000

Inventory: $150,000

Property, plant, and equipment: $400,000

Goodwill: $200,000

Liabilities:

Accounts payable: $75,000

Long-term debt: $250,000

The total fair value of BelAire's net assets is ($50,000 + $100,000 + $150,000 + $400,000 + $200,000) - ($75,000 + $250,000) = $575,000.

The allocation of the consideration paid would be as follows:

Cash paid: $387,500

Fair value of 100,000 shares of Alfonso's common stock: 100,000 shares x $15 per share = $1,500,000

Total consideration paid: $1,887,500

Allocation of consideration:

Cash: $50,000 / $575,000 x $1,887,500 = $164,348

Accounts receivable: $100,000 / $575,000 x $1,887,500 = $328,261

Inventory: $150,000 / $575,000 x $1,887,500 = $492,174

Property, plant, and equipment: $400,000 / $575,000 x $1,887,500 = $1,310,870

Goodwill: $200,000 / $575,000 x $1,887,500 = $657,391

Accounts payable: $75,000 / $575,000 x $1,887,500 = $246,739

Long-term debt: $250,000 / $575,000 x $1,887,500 = $818,478

The entry to record the acquisition would be:

Debit Property, plant, and equipment: $1,310,870

Debit Goodwill: $657,391

Debit Cash: $164,348

Debit Accounts receivable: $328,261

Debit Inventory: $492,174

Credit Accounts payable: $246,739

Credit Long-term debt: $818,478

Credit Cash: $387,500

Credit Common stock: $1,500,000

This entry records the allocation of the total consideration paid to the assets and liabilities acquired, as well as the issuance of Alfonso's common stock to BelAire's shareholders.

The rate of change in the total amount charged with respect to the time spent on the service call is $40 per hour.

The mathematical concept of rate of change describes how much one quantity changes in relation to another. It is calculated as the difference between the change in the output (the dependent variable) and the change in the input (independent variable).

The rate of change, in other words, informs us of how quickly or slowly something is changing through time or in relation to another variable. When tracking the distance an automobile travels over time, for instance, the rate of change would be the car's speed, which is calculated as the distance traveled divided by the time required.

Slope, which is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a graph, is a common way to depict the rate of change. The rate of change of the dependent variable with respect to the independent variable is depicted by the slope of a straight-line graph.

The rate of change in the total amount charged with respect to the time spent on the service call represents the slope of the graph. We can find the slope of the graph by taking any two points on the line and calculating the change in y (total amount) divided by the change in x (time spent on the service call).

Looking at the graph, we can choose two points: (2, 130) and (8, 370). The change in y is:

370 - 130 = 240

And the change in x is:

8 - 2 = 6

Therefore, the slope (rate of change) of the graph is:

240/6 = 40

So the rate of change in the total amount charged with respect to the time spent on the service call is $40 per hour.

Therefore, the correct option is $40 per hour.

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The question you have indicated appears incomplete. However, here is a very similar example that uses the same logic and principles of probability and proportion. In this case, the proof is given below.

a.

Given: f(x) = 2(x+2)/5 , 0 <x< 1

Calculating P(0 <X< 1)

P(0 <X< 1) = ∫ f(x) dx {0.1}

Substitute 2(x+2)/5 for f(x)

P(0 <X< 1) = ∫ 2(x+2)/5 dx {0.1}

P(0 <X< 1) = 2/5 ∫(x+2) dx {0.1}

Integrate with respect to x

P(0 <X< 1) = 2/5 (x²/2+2x) {0.1}

P(0 <X< 1) = 2/5 (1²/2+2(1))

P(0 <X< 1) = 2/5 (½+2)

P(0 <X< 1) = 2/5 * 5/2

P(0 <X< 1) = 1 ----- Proved

b.

Here we're to solve for P(¼<X< ½)

P(¼<X< ½) = ∫ f(x) dx {¼,½}

Substitute 2(x+2)/5 for f(x)

P(¼<X< ½) = ∫ 2(x+2)/5 dx {¼,½}

P(¼<X< ½) = 2/5 ∫(x+2) dx {¼,½}

P(¼<X< ½) = 2/5 (x²/2+2x) {¼,½}

P(¼<X< ½) = 2/5 [ (½²/2+2(½)) - (¼²/2+2(¼))]

P(¼<X< ½) = 2/5 [(⅛+1)-(1/32 + ½)]

P(¼<X< ½) = 0.2375

The probability that more than ¼ but fewer than ½ of the people contacted will respond to this type of solicitation is 0.2375

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Full Question:

The proportion of people who respond to a certain mail-order solicitation is a continuous random variable X that has the density function f(x) = 2(x+2)/5 , 0 <x< 1, 0, elsewhere.

(a) Show that P(0 <X< 1) = 1.

(b) Find the probability that more than ¼ but fewer than ½ of the people contacted will respond to this type of solicitation.