Question

Kara is building raised garden beds for her backyard. The total cost C(x) in dollars is given by C(x) = 125 + 16x, where x is the number of pieces of wood required for the boxes.

a. Find the inverse function C⁻¹(x).
b. If the total cost was $269 and each piece of wood was 12 feet long, how many total feet of wood were used?

Answer 1

The inverse function of C(x) = 125 + 16x is C⁻¹(x) = (x - 125) / 16. For a total cost of $269, Kara used 9 pieces of wood. Since each piece was 12 feet long, she used a total of 108 feet of wood.

Step-by-step explanation:

Kara is building raised garden beds and the total cost C(x) of wood used in dollars is given by the linear equation C(x) = 125 + 16x, where x represents the number of pieces of wood required.

a. Find the inverse function C⁻¹(x).

To find the inverse function, we will solve for x in terms of C(x):

Replace C(x) with y: y = 125 + 16x

Swap x and y: x = 125 + 16y

Solve for y: y = (x - 125) / 16

The inverse function is C⁻¹(x) = (x - 125) / 16.

b. Calculate the total feet of wood used.

The total cost was $269. Using the inverse function:

C⁻¹(269) = (269 - 125) / 16 = 144 / 16 = 9

Thus, Kara used 9 pieces of wood. Since each piece of wood was 12 feet long, the total feet of wood used is 9 pieces × 12 feet/piece = 108 feet.