The function h(t)=-4t^2+14t+8 gives the height h, in feet, of a football as a function of time t, in seconds, after it is kicked. How long does it take for the football to hit the ground?

Answer:

B. w - 1

Step-by-step explanation:

Using PEMDAS, we start see there are no expressions inside parenthesis or exponents we move to multiplication/division. The expression 1/4(4) can be rewritten as 1/4 * 4 or as 4/4, both of which equal 1.

So w - 1 is an equivalent expression.

Answer:

To find (f-g)(x), we need to subtract g(x) from f(x):

(f-g)(x) = f(x) - g(x)

Given f(x) = 2x^2 + 3 and g(x) = x^2 - 7, we can substitute these expressions into the formula above:

(f-g)(x) = f(x) - g(x)

= (2x^2 + 3) - (x^2 - 7)

Now we simplify by combining like terms:

(f-g)(x) = 2x^2 + 3 - x^2 + 7

= x^2 + 10

Therefore, (f-g)(x) = x^2 + 10.

Answer: To find (f-g) (x)

(f-g)(x)= f(x) -g(x)

Given f(x) = 2x^2 + 3 and g(x) = x^2 - 7, we can substitute these expressions into the formula above:

(f-g)(x) = f(x) -g(x)

=(2x^2+3) - (x^2-7)

Now we can simplify the equation:

(f-g)(x)= 2x^2+3 - x^2+7

= x^2 + 10

Hence, (f-g)(x) = x^2+10

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To calculate the new salary amount, we need to add the increase to the original salary.

The increase in salary is 8% of the original salary, which is:

8/100 x K31 600 = K2 528

So, the new salary amount is:

K31 600 + K2 528 = K34 128

Therefore, the new salary amount is K34 128.

To find the new salary after an 8% increment, first find what 8% of the original salary is by multiplying it by 0.08. Add this amount to the original salary to calculate the new salary, which is K34 128.

In order to calculate the new salary after an 8% increment, you first need to figure out how much 8% of the original salary is and then add this to the original salary. So first, remember that 'percent' means 'per hundred', so to calculate 8% of K31 600, you would multiply 31 600 by 0.08 (which is the decimal equivalent of 8%). This gives you K2 528. Then you add this amount to the original salary to get the new salary. Therefore, K31 600 + K2 528 equals to a new salary of K34 128.

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Part 1:

To see why, notice that when x increases by 1, y increases by 2. This corresponds to the coefficient of x in the equation being 2. Also, when x is 0, y is 2, which corresponds to the y-intercept of 2 in the equation. Therefore, the equation that fits the given table is y = 2x + 2.

Part 2:

(a) 2.35 x 10⁻³ can be written in standard form as 0.00235

(b) 2.35 x 10³ can be written in standard form as 2350

Part 3:

A linear equation is an equation where the highest power of the variable is 1. In other words, the graph of a linear equation is always a straight line.

Part 4:

(b) (2,4):

For y=x+3, when x=2, y=5.

For -2x+y=1, when x=2, y=5.

Since (2,4) satisfies both equations, it is a solution.

(c) (2,5):

For y=x+3, when x=2, y=5.

For -2x+y=1, when x=2, y=5.

Since (2,5) satisfies both equations, it is a solution.

Part 5:

Using the Pythagorean Theorem:

a² + b² = c²

8² + b² = 10²

64 + b² = 100

b² = 100 - 64

b² = 36

b = 6

Use the formula for the area of a triangle:

A = (1/2)bh

A = (1/2)(8)(6)

A = 24 cm²

Therefore, the area of triangle ABC is 24 cm². Answer: (b) 24 cm²

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The inverse function is:

f⁻¹(y) = √(y - 2), where y ≥ 2.

An inverse function is a function that "undoes" the action of another function. In other words, if we start with a value, apply a function to it, and then apply the inverse function to the result, we should get back to the original value. For example, if we have a function f(x) that doubles a number, the inverse function would be one that halves a number. The notation for an inverse function is f⁻¹(x), and it is defined as follows:

If f is a function with domain A and range B, then its inverse function f⁻¹         is a function with domain B and range A, where f⁻¹(y) = x if and only if f(x) = y.

To find the inverse of the function f(x) = x² + 2, we need to solve for x in terms of y.

Step 1: Replace f(x) with y.

y = x² + 2

Step 2: Solve for x in terms of y.

y - 2 = x²

±√(y - 2) = x

Note that we use ± because when we take the square root of a number, we get both a positive and a negative solution. However, since the domain of the function is x ≥ 0, we only consider the positive solution.

So the inverse function is:

f⁻¹(y) = √(y - 2), where y ≥ 2.

To sketch the graphs of f(x) and f¹(x) on the same coordinate plane:

Step 1: Plot a few points on the graph of f(x).

We can choose some x-values, plug them into f(x) to find the corresponding y-values, and then plot the points (x, y) on the coordinate plane. For example:

f(0) = 2, so (0, 2) is a point on the graph.

f(1) = 3, so (1, 3) is a point on the graph.

f(2) = 6, so (2, 6) is a point on the graph.

We can also notice that the graph is a parabola that opens upward and has its vertex at (0, 2).

Step 2: Reflect the points across the line y = x to get the graph of the inverse function.

To do this, we swap the x and y coordinates of each point on the graph of f(x) to get the corresponding point on the graph of f⁻¹(x). For example:

(0, 2) becomes (2, 0)

(1, 3) becomes (3, 1)

(2, 6) becomes (6, 2)

We can also notice that the graph of f⁻¹(x) is a curve that starts at (2, 0) and moves upward as x increases.

Step 3: Plot the graphs of both functions on the same coordinate plane.

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Okay, here are the steps to solve this problem:

* Goran used 2 1/2 gallons on Sunday

* Goran used 1/4 gallons on Monday

* So on Sunday he used 2 1/2 gallons and on Monday he used 1/4 gallons

* To find the total gallons used on both days:

** 2 1/2 gallons (used on Sunday)

+ 1/4 gallons (used on Monday)

= 2 3/4 gallons (total used on both days)

So in simplest form as a mixed number, the total gallons Goran used on both days combined is:

2 3/4

[tex]\sf 2\dfrac{3}{4}.[/tex]

To find this answer, all we need to do is add up both of the fractions, that will give us the total amount of gas used on both days. Let's calculate:

[tex]\sf 2\dfrac{1}{2} =\\ \\\dfrac{2}{2}+\dfrac{2}{2}+\dfrac{1}{2}=\dfrac{5}{2}[/tex]

[tex]\sf \dfrac{5}{2}+ \dfrac{1}{4}[/tex]

[tex]\sf \dfrac{5}{2}+ \dfrac{1}{4}= \dfrac{(5*4)+(2*1)}{2*4}= \dfrac{(20)+(2)}{8}=\dfrac{22}{8}=\dfrac{11}{4}[/tex]

[tex]\sf \dfrac{11}{4}=2.75[/tex]

Take the entire part of the decimal number (2) and write it as the whole number on the mixed number. Also, since the fraction has a denominator of 4, a unit of this fraction would be 4/4, then, the 2 units that we're going to express as a whole number would be 8/4. So, subtract 8/4 from 11/4 and express in the following fashion:

[tex]\sf 2(\dfrac{11}{4}-\dfrac{8}{4} )\\ \\\\ \sf 2(\dfrac{3}{4}) \\ \\ \\\ 2\dfrac{3}{4}[/tex]

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Therefore, the general solution to the differential equation is

Y = e⁽ˣ⁾2+X+C) - 1 or Y = -e⁽ˣ⁾2+X+C) - 1

A differential equation is an equation that connects the derivatives of one or more unknown functions. It is an equation that uses the derivatives of a function or functions, in other words. Many physical processes, including the motion of objects under the influence of forces, the movement of fluids, and the spread of disease, are modelled using differential equations in science and engineering. Ordinary differential equations (ODEs) and partial differential equations (PDEs) are the two primary categories of differential equations.

To solve this differential equation using separation of variables, we first need to separate the variables Y and X on opposite sides of the equation:

dY / (Y + 1) = (2X + 1) dX

Following that, we incorporate both sides of the problem:

∫ dY / (Y + 1) = ∫ (2X + 1) dX

The integral on the left side can be evaluated using the substitution

u = Y + 1 and du = dY:

ln|Y + 1| = ∫ dY / (Y + 1) = ln |u| + C1

where C1 is the constant of integration.

The integral on the right side can be evaluated using the power rule of integration:

∫ (2X + 1) dX = X² + X + C2

where C2 is another constant of integration.

Putting these results together gives the general solution to the differential equation:

ln|Y + 1| = X² + X + C

where C = C1 + C2 is the combined constant of integration.

To solve for Y, we exponentiate both sides of the equation:

|Y + 1| = e⁽ˣ⁾2+X+C)

Taking into account the absolute value, we have two cases:

Case 1: Y + 1 = e⁽ˣ⁾2+X+C)

Y = e⁽ˣ⁾2+X+C) - 1

Case 2: Y + 1 = -e⁽ˣ⁾2+X+C)

Y = -e⁽ˣ⁾2+X+C) - 1

Therefore, the general solution to the differential equation DY/DX=(Y+1)(2X+1) is:

Y = e⁽ˣ⁾2+X+C) - 1 or Y = -e⁽ˣ⁾2+X+C) - 1

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(b) The transformation stretches the graph by a factor of 5 describes the effect of this transformation on the graph of the quadratic function along the y-axis.

The transformation from y = 2x² to y = 10x² changes the coefficient of the x² term from 2 to 10. This change in the coefficient affects the vertical scaling of the graph of the quadratic function along the y-axis. When we multiply the entire function by a constant, it causes a vertical stretch or compression of the graph depending on the magnitude of the constant.

In this case, the transformation stretches the graph along the y-axis by a factor of 5 since 10 is 5 times greater than 2. This means that the vertical distance between the points on the graph of the function is now 5 times greater than before the transformation.

Therefore, the correct option is (b) - The transformation stretches the graph by a factor of 5.

Correct Question :

If you transform y = 2x2 into y = 10x2, which option below describes the effect of this transformation on the graph of the quadratic function along the y-axis?

a) The transformation shrinks the graph by a factor of 25.

b) The transformation stretches the graph by a factor of 5.

c) The transformation stretches the graph by a factor of 25.

d) The transformation shrinks the graph by a factor of 5.

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Answer: -3 3/7

Step-by-step explanation:

First, you can convert -1 5/7 to an improper fraction by multiplying the denominator with the whole number, then add the answer to the numerator. Like this:

7 × 1 = 7

7 + 5 = 12

12 will be your numerator and 7 will be your denominator. When you're done converting, bring down the negative sign. The fraction should be:

-12/7

Now, since -1 5/7 is -12/7, we can divide.

To divide, we can use, KCF (Keep, Change, Flip)

First, keep -12/7.

Second, change division to multiplication.

Third, flip 1/2.

Your problem will be set up like:

-12 /7 × 2/1

Now, we can solve. Multiply both the numerator AND the denominator. Since you're multiplying, you don't have to change the denominator to match each other. The answer will be:

-24 / 7

However, this fraction is improper. We can use long division to make this fraction proper.

Divide 24 and 7. You'll get 3, with a remainder of 3.

The remainder will be the new numerator and the answer will be the whole number.

3 - Remainder (Whole Number)

3 - Answer (New Numerator)

7 - Dividend (Original Denominator) Will not be changed

Hence, your final answer is:

-3 3/7

Reply below if you have any questions or concerns.

You're Welcome!

- Nerdworm

[tex]-\dfrac{24}{7} }.[/tex]

[tex]\sf -1\dfrac{5}{7}=\\\\ \\ -1+\dfrac{5}{7}=\\\\ \\ -(\dfrac{7}{7}+\dfrac{5}{7})=\\ \\ \\-(\dfrac{12}{7})[/tex]

[tex]\dfrac{-\dfrac{12}{7} }{\dfrac{1}{2} }[/tex]

[tex]-\dfrac{12}{7} }*\dfrac{2}{1} =\\ \\ \\-\dfrac{12}{7} }*2=\\ \\ \\-\dfrac{12*2}{7} }\\ \\ \\-\dfrac{24}{7} }[/tex]

Answer:

ZR = 145°

Step-by-step explanation:

the secant- secant angle ZAR is half the difference of the measures of the intercepted arcs , that is

∠ ZAR = [tex]\frac{1}{2}[/tex] ( ZR - KV )

30 = [tex]\frac{1}{2}[/tex] (5x + 10 - (3x + 4) ) ← multiply both sides by 2 to clear the fraction

60 = 5x + 10 - 3x - 4

60 = 2x + 6 ( subtract 6 from both sides )

54 = 2x ( divide both sides by 2 )

27 = x

Then

ZR = 5x + 10 = 5(27) + 10 = 135 + 10 = 145°

The area of Regular Decagon is 325 square units.

The formula for the area of a regular polygon is :

Area (A) = [number of sides (n) × length of side (l) × apothem (a)] /2

Apothem Formula is given by:

For the decagon, n = 10 (10 sides) and a = 10

Now, we calculate the length of the decagon:

a = l / 2tan(180/n)°

10 = l / 2tan(180/10)°

10 = l / 2 tan18°

l = 10 × 2 × tan18°

l = 6.50 units

We have to find the area of decagon:

A = [number of sides (n) × length of side (l) × apothem (a)] /2

A = (10 × 6.50 × 10)/2

A = 325 square units

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Foe complete question, to see the attachment.

Answer:

[tex]x = -\sqrt{5}[/tex]

Step-by-step explanation:

[tex]4 = x^2-1\\5 = x^2\\x = \frac{+}{-} \sqrt{5}[/tex]

We would then chose negative square root 5 since it is in quadrant two

A Ferris wheel reaches a maximum height:

(a) sine function for the height of Emma y = 60 sin (2π/720 t)

(b) cosine function for Eva y = 4 + 60 cos (2π/720 t - π/2)

(c) sine function for Matthew y = 32 + 30 sin (2π/720 t - π/2)

(a) Let's start by determining the amplitude and period of the sine function for Emma's height on the Ferris wheel.

The maximum height of the ride is 60 m, which will be the amplitude of the function.

The period is the time it takes for one complete revolution, which is 12 minutes or 720 seconds.

The general form of the sine function is y = A sin (ωt + φ),

where A = amplitude, ω = angular frequency (2π divided by the period), t = time, and φ = phase shift.

So, the sine function for Emma's height can be written as:

y = 60 sin (2π/720 t)

(b) The cosine function for Eva's height will be similar to the sine function for Emma's height, but with a phase shift of 90 degrees, since Eva is starting at the lowest point of the ride (when the sine function is equal to 0).

The general form of the cosine function is y = A cos (ωt + φ), so the cosine function for Eva's height can be written as:

y = 4 + 60 cos (2π/720 t - π/2)

4 meters to account for the height of the staircase.

(c) Matthew is at the same height as the central axle of the wheel, which means his height will vary sinusoidally with the same period as Emma's height, but with a different amplitude and phase shift.

Since his maximum height is halfway between the minimum and maximum heights of the Ferris wheel (i.e. at a height of 32 meters), his amplitude will be half of Emma's amplitude (i.e. 30 meters).

The phase shift will depend on where he is in relation to the starting point of Emma's height function, but assume that he is starting at the lowest point of the ride (like Eva), so his phase shift will also be 90 degrees.

Therefore, the sine function for Matthew's height can be written as:

y = 32 + 30 sin (2π/720 t - π/2)

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The number line where we plotted -1 1/6 and 17/6 is added as an attachment

From the question, we have the following parameters that can be used in our computation:

-1 1/6 and 17/6

To start with, we convert both numbers to the same form

i.e. decimal or fraction

When converted to fractions, we have

-7/6 and 17/6

This means that we can plot -7/6 at -7 and 17/6 at point 17 where the difference in each interval is 1/6

Using the above as a guide, we have the following:

The number line is attached

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It would take approximately 19.94 years to accumulate a total of $10,000 from an initial investment of $3,900 at a simple annual interest rate of 7.84%.

To determine how long it will take to accumulate $10,000 from an initial investment of $3,900 at a simple annual interest rate of 7.84%, we can use the formula for simple interest:

I = Prt

where I is the interest earned, P is the principal or initial investment, r is the annual interest rate as a decimal, and t is the time in years.

To find the time required to reach a final amount of $10,000, we can rearrange the formula:

t = (I/P) / r

First, we need to calculate the interest earned on the initial investment:

I = Prt

I = (3900)(0.0784)t

I = 305.76t

To reach $10,000, we need to earn an additional $10,000 - $3,900 = $6,100 in interest. So we can set up the equation:

6100 = 305.76t

Solving for t, we get:

t = 6100 / 305.76

t ≈ 19.94 years

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Part A

Canine Cakes: $1.8 per pound

Bark Bits: $1.75 per pound

Woofy Waffles: $1.5 per pound

Part B:

Woofy Waffles has the best buy. The unit rate is the least of three.

The term "rate" generally refers to the measure of change in one quantity with respect to another quantity. It describes how one quantity changes in relation to another quantity, often expressed as a ratio or a fraction.

The buying price, also known as the purchase price or the cost price, is the amount of money that is required to purchase an item, product, or service. It is the price at which a buyer acquires a product or service from a seller or a vendor.

Canine cakes= 30pounds

cost =$54

Unit rate= $54/30 = $1.8 per pound

Bark Bits= 40 pounds

cost =$70

Unit rate= $70/40= $1.75 per pound

Woofy Waffles = 28pounds

cost =$42

Unit rate=$42/28= $1.5 per pound

The best buy is 28 pounds of Woofy Waffles for $42. The rate is least of the three.

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Based on the above, the total  Card area is C: 30 square cm.

Note that the shape is congruent to the one shown on the grid, and thus all shape has the same area. So, to find the total area that is covered by 5 shapes, we need to find the area of one shape as well as then multiply it by 5.

To find total area covered by 5 congruent shapes, we need to multiply one shape's area by 5. To find the area, count 6 unit squares covered by shape on the grid.

Therefore, the total Card area covered by 5 shapes is:

5 × 6 cm² = 30 cm²

Therefore, the total area is C: 30 square cm.

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The solution to the system of equations is x = -9; y = 15 and z = -4

An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.

Given the system of equations:

-3x - y + z = 8     (1)

-3x - y + 3z = 0   (2)

and:

x - 3z = 3            (3)

Solving the three equations simultaneously gives:

x = -9; y = 15 and z = -4

The solution is x = -9; y = 15 and z = -4

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There is a probability of 0.117 for at least 2 of the tiles to be vowel tiles.

Given that,

Probability that the tiles getting is a vowel tile is 30%.

P(V) = 30% = 0.3

That is there will be only 3 tiles out of 10 tiles which are vowels.

Out of 8, there will be 8 × 0.3 = 2.4 tiles which are vowels.

There would be either 2 or 3 vowels.

Probability that at least 2 of them is vowel tiles is,

Probability = (0.3)² + (0.3)³

                  = 0.117

Here,

The simulation which can be used to fairly represent the situation is,

Use a computer to randomly generate 8 numbers from 1 to 10. Each time 1, 2 or 3 appears, it represents a vowel tile.

Hence the required probability is 0.117.

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Answer: 71.4%

Step-by-step explanation:

Mean = ~5 mins

Deviation = 3 mins

This means that you have a range of 2 mins - 8 mins.

2, 3 || 4, 5, 6, 7, 8

The double lines represent 3 minutes or less, and more than 3 minutes.

There are 7 number, 5 are greater than 3, that is 71.4%.

Answer: 5.36%

Step-by-step explanation:

The formula for the effective annual yield (EAR) when the interest is compounded daily is given by:

(1 + r/365)^365 - 1

where r is the annual interest rate.

In this case, the annual interest rate is 5.2% or 0.052. Substituting this value into the formula, we get:

(1 + 0.052/365)^365 - 1 = 0.0536

Multiplying this value by 100 gives the effective annual yield as a percentage:

0.0536 x 100 = 5.36%

Therefore, the effective annual yield of the account is 5.36%.

Okay, let's break this down step-by-step:

* The business has $40,000 total to spend on advertising

* Some will go to TV (x), some to radio (y), and some to newspapers (z)

* 3 times as much will go to TV as radio, so x = 3y

* They will spend $8,000 less on radio than newspapers, so y = z - 8,000

* We have:

x = 3y (1)

y = z - 8,000 (2)

x + y + z = 40,000 (3)

To solve this using matrix inversion:

1) Turn the equations into a matrix:

3 1 0 y

1 -1 1 z

1 1 1 x

2 0 40,000

2) Invert the matrix:

0.3333 0.3333 0.3333

-0.25 0.75 0

0.125 0.125 0.750

3) Plug in the values from (2) and (3):

y = 0.3333(z - 8,000)

x = 0.125z + 0.125(40,000 - z)

4) Solve for z, the amount for newspapers. We get:

z = 40,000 * (0.75) = 30,000

5) Plug z = 30,000 back into the other equations:

x = 0.125 * 30,000 + 0.125 * 10,000 = 12,000

y = 0.3333 * (30,000 - 8,000) = 8,000

z = 30,000

So in total:

TV (x) = $12,000

Radio (y) = $8,000

Newspapers (z) = $30,000

Does this make sense? Let me know if you have any other questions!

a) The rule that represents the height as a function of the number of bounces is described by H(n) = 1.5 · (71 / 100)ˣ⁻¹. (Correct choice: B)

b) The height at the top of the sixth path is equal to 0.27 meters. (Correct choice: B)

In this problem we need to derive the equation that represents maximum height as a function of the number of bounces:

H(n) = a · (r / 100)ˣ⁻¹

Where:

If we know that a = 1.5 m and r = 71, then the rule for the sequence is:

H(n) = 1.5 · (71 / 100)ˣ⁻¹

And the height at the top of the sixth path:

H(6) = 1.5 · (71 / 100)⁶⁻¹

H(6) = 0.27 m

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Therefore, the equation that cannot be the other equation is A. y = -2x.

An equation is a mathematical statement that shows the equality of two expressions. It usually consists of two sides separated by an equal sign (=). The expressions on both sides of the equal sign can include numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

Here,

To check which equation cannot be the other equation, we can substitute the coordinates of (-1,5) into each of the equations and see if they hold true.

A. y = -2x

When x = -1, then y = -2(-1) = 2, so (-1,5) does not satisfy this equation. Therefore, this cannot be the other equation.

B. y = -5x

When x = -1, then y = -5(-1) = 5, so (-1,5) does satisfy this equation.

C. y = -x + 4

When x = -1, then y = -(-1) + 4 = 5, so (-1,5) does satisfy this equation.

D. y = -x²/2 + 9/2

When x = -1, then y = -(-1)²/2 + 9/2 = 5, so (-1,5) does satisfy this equation.

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Answer: Adam's base pay

Step-by-step explanation:

The equation, y = 0.28x + 38000, is the equation Adam is using to find his annual salary which is said to include base pay and commission pay.

His total commission pay is going to be dependent on his number of sales, which we're told is x.

So, his commission pay is 0.28x.

Therefore, his base pay has to be 38,000; his annual salary if he makes no sales.

The equation expressing the total cost in terms of the number of hours required to accomplish the task is C = 32h + 31.50

Let C be the entire cost of the job, and h represent the number of hours needed to accomplish the job.

The hourly charge for services is K32, hence the total cost for services is 32h.

There is a one-time fee of K31.50, making the total cost:

C = 32h + 31.50

Hence, the equation expressing the total cost in terms of the number of hours required to accomplish the task is C = 32h + 31.50

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Answer:

50.8

Step-by-step explanation:

Answer:

y ≈ 34

Step-by-step explanation:

using the tangent ratio in the right triangle

tan y = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{32}{48}[/tex] = [tex]\frac{2}{3}[/tex] , then

y = [tex]tan^{-1}[/tex] ( [tex]\frac{2}{3}[/tex] ) ≈ 34 ( to the nearest whole number )

answer

y = 33.69°

tan= opposite /adjacent

tan y° = 32 / 48

tan y° = 2/3

tan y° = 0.67

y° = tan^-1 (0.67)

y° = 33.69°