What is the measure of an angle that goes through 2/8 of a circle?

If you provide me with a specific circuit diagram and the relevant details, I would be happy to help you determine the probability of current flowing from s to t when the switch is thrown.

I apologize, but it seems that the circuit diagram you mentioned is not displayed here. Without the circuit diagram, it is not possible for me to provide a specific answer to your question.

However, in general, the probability of current flowing from s to t in an electrical circuit depends on several factors such as the voltage level, the resistance of the circuit components, and the state of the switches. If the switches are all closed, then the probability of current flowing from s to t will depend on the overall resistance of the circuit.

If you provide me with a specific circuit diagram and the relevant details, I would be happy to help you determine the probability of current flowing from s to t when the switch is thrown.

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Answer:

since 2x = z

replace z with 2x

900000 = x+y+z

900000 = x+y+2x

900000 = 3x+y - eqn (1)

79750= 0.08x +0.09y+0.01z

79750 = 0.08x +0.09y+0.01(2x)

79750 = 0.08x+0.09y+0.02x

79750 = 0.10x +0.09y - eqn(2)

from eqn(1)

900000 = 3x + y

y = 900000-3x - eqn(3)

substitute eqn(3) in eqn(2)

79750 = 0.1x +0.09y

79750=0.1x + 0.09(900000-3x)

79750=0.1x+ 81000 - 0.27x

collect like terms

79750 -81000 = 0.1x-0.27x

-1250 = -0.17x

to find x divide both sides by -0.17

x = -1250/-0.17 ~= 7353

since 2x = z

2*7353 = 14706

in eqn(3)

y = 900000-3x

y= 900000-3(7353)

y = 900000-22059

y = 877941

x =7353,y= 877941,z=14706

The quadratic equation where the squares had been completed is:

(x + 2)² = 27/5

Remember the perfect square trinomial:

(a + b)² = a² + 2ab + b²

now we have the quadratic equation:

5x² + 20x - 7 = 0

If we divide it all by 5, we will get.

x² + 4x - 7/5 = 0

Now we can rewrite this as:

(x² + 2*2*x )  - 7/5 = 0

Now we need to add 2² in both sides, we will get:

(x² + 2*2x + 2²) - 7/5 = 2²

(x + 2)² = 4 + 7/5

(x + 2)² = 27/5

There the square is completed.

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The first derivative of S is S' = 1/15.
The second derivative of S is S'' = 0.

To find the first derivative (S'):

Starting with the given equation S = 15 + 344 - 1 15, we can simplify it to S = 344 + 15.

We can take the derivative of each term separately since they are added together.

The derivative of a constant (15 and 344) is always 0, so we only need to take the derivative of 1/15.

S' = d/dx (344 + 15)

= d/dx (359)

= 0 + 0 + (d/dx (1/15))

= 1/15

Therefore, the first derivative of S is S' = 1/15.

To find the second derivative (S''):

We need to take the derivative of the first derivative (S').

Since the derivative of a constant is always 0,

we only need to take the derivative of 1/15.

S'' = d/dx (S')

= d/dx (1/15)

= 0

Therefore, the second derivative of S is S'' = 0.

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True percentage of all American believes that their children have higher standard of living with confidence interval of 95% is between 60% and 66% .

CI is the confidence interval

Answered in the affirmative =  63%

p is the sample proportion =0.63

z is the critical value from the standard normal distribution at the desired confidence level

Using attached z-score table,

95% confidence level corresponds to z=1.96

n is the sample size

Use the margin of error ,

Calculate a confidence interval for percentage of American adults who believe their children will have a higher standard of living.

A margin of error of plus or minus 3% means ,

95% confident that the true percentage falls within 3% of the sample percentage.

Using the formula for a confidence interval for a population proportion,

CI = p ± z×√(p(1-p)/n)

Plugging in the values, we get,

⇒ CI = 0.63 ± 1.96√(0.63(1-0.63)/n)

Solving for n, we get,

n = (1.96/0.03)^2 × 0.63(1-0.63)

⇒ n = 994.87

Rounding up to the nearest whole number, sample size of at least 995.

⇒ CI = 0.63 ± 1.96√(0.63(1-0.63)/995)

⇒CI = 0.63 ± 0.02999

95% confidence interval for the true percentage is,

⇒CI = 0.63 ± 0.03

⇒CI = (0.60, 0.66)

Therefore, 95% confidence interval that between 60% and 66% of all American adults with children believe that their children will have a higher standard of living.

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A ten-by-ten grid has 7 columns shaded. All of the following that represent the part of the grid that is shaded are : The correct answer is (A) 70 and (B) 7.

The information given in the problem tells us that a ten-by-ten grid has 7 columns shaded. Since there are a total of 10 columns in the grid, this means that 7/10 of the columns are shaded.

To express this as a percentage, we can divide 7 by 10 and multiply by 100:

(7/10) x 100 = 70%

Therefore, 70 represents the percentage of columns that are shaded in the grid. Option (A) is correct.

Alternatively, we can express the same proportion as a decimal by dividing 7 by 10:

7/10 = 0.7

Therefore, 0.7 represents the proportion of columns that are shaded in the grid. Option (E) is incorrect because it shows 0.7 as a fraction instead of a decimal.

Option (B) is also correct because it correctly identifies the number of shaded columns as 7. Option (C) is incorrect because it includes both the percentage and the number of shaded columns, which is redundant. Option (D) is incorrect because it shows the proportion of shaded columns as a decimal with an extra zero.

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Noah's estimate of 7.5 hours is closest to the actual mean number of hours of sleep reported by the ten sixth-grade students.

We have,

Compare the estimates given by Lin, Noah, and Diego.

Lin's estimate is 8.5 hours.

Noah's estimate is 7.5 hours.

Diego's estimate is 6.5 hours.

The average of these three estimates:

(8.5 + 7.5 + 6.5) / 3

= 22.5 / 3

= 7.5

It appears that Noah's estimate of 7.5 hours is the closest.

Therefore,

Noah's estimate of 7.5 hours is closest to the actual mean number of hours of sleep reported by the ten sixth-grade students.

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The complete question:

Which estimate of the mean number of hours of sleep reported by the ten sixth-grade students is closest to the actual mean: Lin's estimate of 8.5 hours, Noah's estimate of 7.5 hours, or Diego's estimate of 6.5 hours?

The pole is approximately 3.86 feet tall.

Let's denote the height of the pole as "x" (in feet). From the problem, we know that the pole makes a 75° angle with the road surface, which means that the angle between the pole and the vertical is 90° - 75° = 15°.

Now, we can use the tangent function to find the height of the pole:

tan(15°) = x/15

Multiplying both sides by 15, we get:

x = 15 tan(15°) ≈ 3.86 feet

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In the last 215 days, builders have completed 700 m2 of the alligator habitat that will eventually be 1,200 m2.
It will take approximately 153 days to complete the remaining part of the alligator habitat.

Determine how much longer it will take to complete the alligator habitat, first, we need to find the rate at which the builders are working.
Calculate the work rate
The builders have completed 700 m2 of the 1,200 m2 alligator habitat in 215 days.
Work rate = (completed work) / (number of days)
Work rate = 700 m2 / 215 days = 3.26 m2/day (approximately)
Calculate the remaining work
The total area of the alligator habitat is 1,200 m2, and 700 m2 has been completed.
Remaining work = Total area - Completed work
Remaining work = 1,200 m2 - 700 m2 = 500 m2
Calculate the time to complete the remaining work
Time to complete = (remaining work) / (work rate)
Time to complete = 500 m2 / 3.26 m2/day ≈ 153.37 days
It will take approximately 153 days to complete the remaining part of the alligator habitat.

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It will take approximately 394 more days to complete the alligator habitat.

We can start by finding the proportion of the habitat that has already been completed:

proportion completed = 700 m^2 / 1200 m^2 = 0.5833

This means that there is still 1 - 0.5833 = 0.4167 (or 41.67%) of the habitat left to complete.

Next, we can use a proportion to find out how long it will take to complete the remaining 41.67% of the habitat:

215 days / 0.5833 = x days / 0.4167

Solving for x, we get:

x = 215 days * 0.4167 / 0.5833 ≈ 153 days

Therefore, the total time it will take to complete the alligator habitat is approximately 215 + 153 = 368 days, or about 394 more days from the start.

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In the given circle, measure of angle m is 44° and the measure of angle n is 39°. Thus, the value of m is 44 and the value of n is 39

From the question, we are to determine the values of m and n in the given circle

From one of the circle theorems, we have that

The angles at the circumference subtended by the same arc are equal. That is, angles in the same segment are equal.

In the given diagram,

Angle m is in the same segment as the angle that measures 44°

Since angles in the same segment are equal,

Measure of angle m = 44°

Also,

Angle n is in the same segment as the angle that measures 39°

Since angles in the same segment are equal,

Measure of angle n = 39°

Hence,

m ∠m = 44°

m ∠n = 39°

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To find the extremes of 4x−4y subject to the condition x2 + 2y2 = 1, we can use the method of Lagrange multipliers.

First, we set up the Lagrange equation:

∇f(x,y) = λ∇g(x,y)

where f(x,y) = 4x-4y and g(x,y) = x2 + 2y2 - 1.

Taking partial derivatives, we have:

∂f/∂x = 4
∂f/∂y = -4
∂g/∂x = 2x
∂g/∂y = 4y

Setting these equal to their respective Lagrange multipliers, we have:

4 = 2λx
-4 = 4λy
x2 + 2y2 = 1

Solving for x and y in terms of λ, we get:

x = 2λ/4 = λ/2
y = -λ/4

Substituting these back into the constraint equation, we have:

(λ/2)2 + 2(-λ/4)2 = 1
λ2/4 + λ2/8 = 1
3λ2/8 = 1
λ2 = 8/3

Taking the positive and negative square roots of λ2, we have:

λ = ±2√2/3

Substituting these values back into x and y, we get:

For λ = 2√2/3:
x = (2√2/3)/2 = √2/3
y = -(2√2/3)/4 = -√2/6

For λ = -2√2/3:
x = (-2√2/3)/2 = -√2/3
y = -(-2√2/3)/4 = √2/6

Now we can find the extreme values of f(x,y) by plugging in these values of x and y:

f(√2/3, -√2/6) = 4(√2/3) - 4(-√2/6) = 4√2
f(-√2/3, √2/6) = 4(-√2/3) - 4(√2/6) = -4√2

Therefore, the maximum value of 4x-4y subject to the condition x2 + 2y2 = 1 is 4√2 and the minimum value is -4√2.

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The product of the expression 5x(x²) is 5x³.

1. Write down the given expression: 5x(x²)
2. Apply the distributive property, which states that a(b + c) = ab + ac. In this case, we have a single term inside the parentheses, so the expression becomes: 5x * x²
3. Multiply the coefficients (numbers) together: 5 * 1 = 5
4. Multiply the variables together, which means adding the exponents since they have the same base (x): x¹* x² = x⁽¹⁺²⁾ = x³
5. Combine the result from steps 3 and 4: 5x³

The product of the expression 5x(x²) can be found by multiplying the coefficients (numbers) and adding the exponents of the variables (letters). In this case, we have 5 times x times x squared.

5 times x equals 5x, and x squared means x times x, so we can rewrite the expression as:

5x(x²) = 5x(x*x) = 5x³

So, the product of the expression 5x(x²) is 5x³.

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One point that would lie on the second line is (0,-4). Another  possible point on the 2nd line is (0, 12).

To find the equation of the first line, we can use the slope-intercept form:

y = mx + b

where m is the slope and b is the y-intercept. The slope of the line passing through (-3,8) and (1,9) can be found using the formula:

m = (y2 - y1) / (x2 - x1)

m = (9 - 8) / (1 - (-3))

m = 1/4

Using one of the points and the slope, we can find the y-intercept:

8 = (1/4)(-3) + b

b = 9

So the equation of the first line is:

y = (1/4)x + 9

To find the equation of the second line, we need to use the fact that it is perpendicular to the first line. The slopes of perpendicular lines are negative reciprocals, so the slope of the second line is:

m2 = -1/m1 = -1/(1/4) = -4

Using the point-slope form, we can write the equation of the second line:

y - 4 = -4(x + 2)

y - 4 = -4x - 8

y = -4x - 4

To find a point that lies on this line, we can plug in a value for x and solve for y. For example, if we let x = 0, then:

y = -4(0) - 4

y = -4

So the point (0,-4) lies on the second line.

Therefore, another point that would lie on the second line is (0,-4).

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The loan with the smallest monthly payment is the 30-month loan at 6% annual simple interest rate, with a monthly payment of $45.83.

To determine the loan with the smallest monthly payment, we need to calculate the monthly payment for each loan option and compare them.

We can use the formula for monthly payment on a simple interest loan:

monthly payment = (principal + (principal * interest rate * time)) / total number of payments

where:

We can compute the monthly payments for each loan choice using this formula:

1. 12 Monthly interest rate = 0.0625/12 = 0.00521, monthly payment = (1250 + (1250 * 0.00521 * 12)) / 12 = $107.35

2. 18 months at 6.75%: monthly interest rate = 0.0675/12 = 0.00563, monthly payment = (1250 + (1250 * 0.00563 * 18)) / 18 = $81.96

3. 24 months at 6.5%: monthly interest rate = 0.065/12 = 0.00542, monthly payment = (1250 + (1250 * 0.00542 * 24)) / 24 = $66.14

4. 30 months at 6%: monthly interest rate = 0.06/12 = 0.005, monthly payment = (1250 + (1250 * 0.005 * 30)) / 30 = $45.83

Based on these calculations, the loan with the smallest monthly payment is the 30-month loan at 6% annual simple interest rate, with a monthly payment of $45.83.

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The length of the rectangle is 19 feet and its width is 13 feet.

Let's denote the width of the rectangle by w. Then, according to the problem statement, the length of the rectangle is 6 feet longer, which means it is equal to w + 6.

The perimeter of a rectangle is given by the formula:

perimeter = 2 × length + 2 × width

Substituting the expressions for length and width that we have just found, we get:

64 = 2 × (w + 6) + 2w

Simplifying the right-hand side:

64 = 2w + 12 + 2w

64 = 4w + 12

52 = 4w

w = 13

So the width of the rectangle is 13 feet. Using the expression for the length we found earlier, the length is:

length = w + 6 = 13 + 6 = 19

Therefore, the length of the rectangle is 19 feet and its width is 13 feet.

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The value of angle x = 114°.

From the figure, it is clear that The interior angle of a triangle is 39°, by the law of opposite angle.

The sum of the interior angle of a triangle is 180°

37° + 39° + ∠unknown1 = 180°

∠unkonown1 = 180° - 37° - 39°

∠unknown1 = 104°

The sum of the exterior angle and the interior angle is 180°.

∠unknown2+ ∠unknown 1= 180°

∠unknown2 = 180° - 104°

∠unknown2 = 76°

The sum of the interior angle of a triangle is 180°

∠unknown3 + ∠unknown2 + 38 = 180

∠unknown3 + 76° + 38 = 180

∠unknown3= 66°

The sum of the exterior angle and the interior angle is 180°.

∠X + <unknown3 = 180°

∠X = 180° - 66°

∠X = 114°

The value of the angle x is 114°.

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If a man bought a diamond for $32 in 1682 and the man had instead put the $32 in the bank at 3% interest compounded continuously, then the value of the diamond in 2003 would be $554,311.

The given problem is related to exponential growth. In this problem, the continuous compounding formula will be used to find the value of $32 in 2003.

The formula for continuous compounding is given by:

A = Pert   Where,

P is the principal amount,

r is the annual interest rate,

e is the Euler's number which is approximately 2.71828, and

t is the time in years.

Using the formula, we get:

A = 32e^(0.03 x 321)

A = 32e^9.63

A = 32 x 17322.23

A = $ 554311.36

Thus, $32 invested at 3% compounded continuously from 1682 to 2003 would be worth $554,311.

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The solution to the given equation is x = -3/4 or x = 1.

In order to find the solution, we start by squaring both sides of the equation to eliminate the square roots:

12x² - 4 = 4x + 8

Next, we simplify the equation by moving all the terms to one side:

12x² - 4x - 12 = 0

Now we can factor the quadratic equation:

4x² - x - 3 = 0

By factoring or using the quadratic formula, we find that the equation can be written as:

(4x + 3)(x - 1) = 0

Setting each factor equal to zero gives us the solutions:

4x + 3 = 0 or x - 1 = 0

Solving for x in each equation yields:

x = -3/4 or x = 1

Therefore, the solution to the given equation is x = -3/4 or x = 1.

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The Laplace transform of the solution to the initial value problem y'' + y' - 2y = 0, y(0) = -5, is Y(s) = (5s + 4) / (s² + s - 2), and Y(5) = 29 / 28.

To find the Laplace transform of the solution to the initial value problem y'' + y' - 2y = 0, y(0) = -5, we can apply the Laplace transform to both sides of the differential equation and use the initial condition to solve for the Laplace transform of y.

Taking the Laplace transform of both sides of the differential equation, using the linearity and derivative properties of the Laplace transform, we get:

L{y'' + y' - 2y} = L{0}

s² Y(s) - s y(0) - y'(0) + s Y(s) - y(0) - 2 Y(s) = 0

s² Y(s) - 5s + s Y(s) + 4 + 2 Y(s) = 0

Simplifying and solving for Y(s), we get:

Y(s) = (5s + 4) / (s²+ s - 2)

To find Y(5), we substitute s = 5 into the expression for Y(s):

Y(5) = (5(5) + 4) / ((5)² + 5 - 2)

Y(5) = 29 / 28

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first off, let's look at the equation of the circle

[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{}{h}~~,~~\underset{}{k})}\qquad \stackrel{radius}{\underset{}{r}} \\\\[-0.35em] ~\dotfill\\\\ (x-\stackrel{h}{3})^2+(y-\stackrel{k}{7})=169\implies (x-\stackrel{h}{3})^2+(y-\stackrel{k}{7})=\stackrel{ r }{13^2}[/tex]

so we have a circle centered at (3 , 7) with a radius of 13, Check the picture below.

so the line we want is the line in purple, which is tangential to the circle and therefore perpendicular to the blue line.

keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the blue line

[tex](\stackrel{x_1}{3}~,~\stackrel{y_1}{7})\qquad (\stackrel{x_2}{15}~,~\stackrel{y_2}{2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{2}-\stackrel{y1}{7}}}{\underset{\textit{\large run}} {\underset{x_2}{15}-\underset{x_1}{3}}} \implies \cfrac{ -5 }{ 12 } \implies - \cfrac{5 }{ 12 } \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{-5}{12}} ~\hfill \stackrel{reciprocal}{\cfrac{12}{-5}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{12}{-5} \implies \cfrac{12}{ 5 }}}[/tex]

so we're really looking for the equation of a line whose slope is 12/5 and it passes through (15 , 2)

[tex](\stackrel{x_1}{15}~,~\stackrel{y_1}{2})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{12}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{ \cfrac{12}{5}}(x-\stackrel{x_1}{15}) \\\\\\ y-2=\cfrac{12}{5}x-36\implies {\Large \begin{array}{llll} y=\cfrac{12}{5}x-34 \end{array}}[/tex]

The rule that best represents the dilation that was applied to quadrilateral FGHJ to create quadrilateral F'G'H'J' is option B, which is (x, y) à (1.4x, 1.4y).

A dilation is a transformation that changes the size of an object without changing its shape. It is performed by multiplying the coordinates of each point by a scale factor.

In this case, the center of dilation is the origin, which means that the coordinates of each point are multiplied by the same scale factor in both the x and y directions.

The scale factor can be found by comparing the corresponding side lengths of the two quadrilaterals. In this case, the scale factor is 1.4, which means that the lengths of the sides of F'G'H'J' are 1.4 times the lengths of the corresponding sides of FGHJ.

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Answer:

  A)  x < 15

Step-by-step explanation:

You want the solution to x/3 < 5.

The steps to solving an inequality are basically identical to the steps for solving an equation. There are a couple of differences:

If this were and equation, it would be a "one-step" equation. That step is to multiply both sides by the inverse of the coefficient of x.

The coefficient of x is 1/3. Its inverse is 3. Multiplying both sides by 3, we have ...

  3(x/3) < 3(5)

  x < 15 . . . . . . . . . simplify

Note that 3 is a positive number, so we leave the inequality symbol pointing the same direction.

__

Additional comment

We can swap the sides of an equation based on the symmetric property of equality:

  a = b   ⇔   b = a

When we swap the sides of an inequality, we need to preserve the relationship between them. (This is the meaning of "respect the direction of the inequality symbol".)

  a < b   ⇔   b > a

Besides multiplying and dividing by a negative number, there are other operations that affect the order of values.

Note that the 1/x function is another one that has negative slope, which is why it reverses the ordering for values with the same sign. (It has no effect on ordering of values with opposite signs.)

In a regular tiling, if there are six polygons meeting at a vertex, then the angles at the vertex are 120 degrees.

This is because each regular polygon has interior angles that are multiples of 180 degrees divided by the number of sides. For a regular hexagon, which has six sides, each interior angle measures 120 degrees. When six regular hexagons meet at a vertex in a regular tiling, the total angle sum at the vertex is 720 degrees (6 times 120 degrees).

Since the angles must be divided equally among the six hexagons, each angle at the vertex is 120 degrees.

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Answer:

As x approaches ∞, y approaches -∞

As x approaches -2.5, y approaches ∞

Step-by-step explanation:

Answer and Explanation:

The height of the small bottle can be calculated using the ratio of the width of the large and small bottles.

Ratio of width = Large bottle width / Small bottle width

Ratio of width = 9.9 cm / 4.5 cm

Ratio of width = 2.2

Therefore, the height of the small bottle can be calculated by multiplying the ratio of width with the height of the large bottle.

Height of small bottle = Ratio of width x Height of large bottle

Height of small bottle = 2.2 x 19.8 cm

Height of small bottle = 43.56 cm

Answer:

Step-by-step explanation:

1807

(a) The cost of City Bus for driving 10 miles = $3.

(b) The cost of City Bus for driving 25 miles = $7.5.

(c) The cost of City Bus for driving 42 miles = $12.6.

(d) The cost of City Bus for driving 68 miles = $20.4.

We previously choose City Bus as our mode transport since the per mile cost for City Bus is less.

Let the model for City Bus be f(x) = cx + d, where f(x) is total cost and x is number of miles.

From the table of Taxi we get, f(2) = 0.60; f(4) = 1.20; f(6) = 1.80 and f(8) = 2.40.

So, 2a + b = 0.60 and 4a + b = 1.20

(4a + b) - (2a + b) = 1.20 - 0.60

2a = 0.60

a = 0.60/2 = 0.30

Now, f(8) = 2.40

8*0.30 + b = 2.40

2.40 + b = 2.40

b = 2.40 - 2.40 = 0

So the function rule for City Bus is, f(x) = 0.3x.

(a) Total cost to drive 10 miles is,

f(10) = 0.3*10 = 3

(b) Total cost to drive 25 miles is,

f(25) = 0.3*25 = 7.5

(c) Total cost to drive 42 miles is,

f(42) = 0.3*42 = 12.6

(d) Total cost to drive 68 miles is,

f(68) = 0.3*68 = 20.4

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The equation of the two straight lines through the origin making an angle of 45° with the line px + qy + r = 0 is (p²-q)(x² - y²) + 4pqxy = 0.

To prove that the equation of the two straight lines through the origin, making an angle of 45° with the line px + qy + r = 0, is given by (p²-q)(x² - y²) + 4pqxy = 0, we can use the concept of slopes and trigonometric identities.

Let's consider the line px + qy + r = 0. The slope of this line is given by -p/q.

Now, the lines making an angle of 45° with this line will have slopes equal to tan(45°), which is 1.

Using the formula for the tangent of the sum of angles, we have:

tan(45°) = (m - (-p/q))/(1 + m(-p/q)), where m represents the slope of one of the lines.

Simplifying the equation, we get:

1 = (mq + p)/(q - mp)

Cross-multiplying and rearranging the terms, we obtain:

(p² - q)(m² - 1) + 2pqm = 0

Since these lines pass through the origin (0,0), we can replace m with y/x. Substituting y/x for m in the equation above, we get:

(p² - q)(x² - y²) + 2pqxy = 0

Further simplifying the equation, we arrive at:

(p² - q)(x² - y²) + 4pqxy = 0

Hence, we have proven that the equation of the two straight lines through the origin, making an angle of 45° with the line px + qy + r = 0, is given by (p²-q)(x² - y²) + 4pqxy = 0.

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Answer:

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The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n:

As per above mentioned definition we see that:

Hence the quotient of (n + 3)! and (n + 1)! is: