SOLUTION: Bethany Link delivers parts for several local auto parts stores. She charges clients $.80 per mile driven. She has determined that if she drives 2,350 miles in a month, her average

Question 950160: Bethany Link delivers parts for several local auto parts stores. She charges clients $.80 per mile driven. She has determined that if she drives 2,350 miles in a month, her average operating cost is $.76 per mile. If Bethany drives 3,350 miles in a month, her average operating cost is $.56 per mile.
Variable Cost per Unit?
Fixed Cost?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
she charges .80 per mile.

that's her revenue.

her costs have a fixed component and a per mile component.

her cost formula would be c = f + v * m

c/m * m is the total cost.

it's the cost per mile * the number of miles.

you'll see why this is shown this way as we go through the specific formulas below.

f is the fixed cost.

v is the variable cost per mile.

m is the number of miles.
She has determined that if she drives 2,350 miles in a month, her average operating cost is $.76 per mile.

c/m is equal to .76

c is equal to c/m * m which is equal to .76 * 2350 = 1786.

the formula of c = f + v * m becomes:

1786 = f + 2350 * v

If Bethany drives 3,350 miles in a month, her average operating cost is $.56 per mile.

c/m = .56

c = c/m * m which is equal to .56 * 3350 = 1876

the formula of c = f + v*m becomes:

1876 = f + 3350 * v

she has 2 formulas to work with:

first formula is:

1786 = f + 2350 * v

second formula is:

1876 = f + 3350 * v

these 2 formulas need to be solved simultaneously so that she can get a common solution for f and v that applies equally to both of them.

start with:

1786 = f + 2350 * v (first equation)
1876 = f + 3350 * v (second equation)

subtract the first equation from the second equation to get:

90 = 1000 * v

solve for v to get:

v = 90 / 1000 = .09

replace v in either equation with .09 and solve for f.

we'll use the first equation, even though the second equation would work just as well.

you will get the same answer either way as long as you calculated v correctly.

start with:

1786 = f + 2350 * v (first equation)

replace v with .09 to get:

1786 = f + 2350 * .09

simplify to get:

1786 = f + 211.5

solve for f to get f = 1786 - 211.5 = 1574.5

that's the fixed cost.

the fixed cost is 1574.5
the variable cost is .09 per mile.

apply to both equations to see if they are true.

equations of:

1786 = f + 2350 * v (first equation)
1876 = f + 3350 * v (second equation)

become:

1786 = 1574.5 + 2350 * .09 (first equation)
1876 = 1574.5 + 3350 * .09 (second equation)

evaluate both equations to get:

1786 = 1786
1876 = 1876

this confirms the solutions are correct and apply equally to both equations.

her fixed cost is 1574.5 and her variable cost is .09 per mile.

since she charges 80 cents per mile, she makes more profit on the long runs than on the short runs.

this is because her average cost per mile is lower on the long runs than on the short runs.

this has to do with the relationship between her fixed costs and her variable costs.

lower fixed costs and the relative costs would have been closer to each other.

higher fixed costs and the relative costs would have been further apart from each other.

the wrinkle in this problem is that they gave you the cost in terms of an average cost per mile.

you needed to translate that to a total cost by multiplying it by the miles in order for the formula to work.

working back, when you find the total cost, you need to divide it by the number of miles to get the total cost per mile.

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you can stop here because you already have the solution to your problem.

that solution is that the fixed cost is 1574.5 and the variable cost is .09 per mile.

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if you want to go further for some additional information, you may, but you don't have to.

with the information that you have, you can calculate her break even point.

that would be the point where her cost is equal to her revenue.

use the equation .80 * x = 1574.5 + .09 * x

x is the number of miles in this equation.

solve for x to get x = 2217.605634 miles.

if you plug that value in the formula, you would get:

.80 * x = 1574.5 + .09 * x would become:

.80 * 2217.605634 = 1574.5 + .09 * 2217.605634 which would become:

1774.084507 = 1774.084507

this confirms that the break even point is at 2217.605634 miles.

at that distance she would break even.

at greater than that distance she makes money.

at less than that distance she loses money.

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