Question

A company produces custom greeting cards. The cost c (in dollars) of producing n greeting cards per month can be modeled by the function c = 500 * 1.25n. Find the inverse of the model.

Answer 1

The inverse of the cost model function c = 500 × 1.25^n is found by swapping c and n, isolating the exponential term, applying logarithms, and solving for c, resulting in c = log(n/500) / log(1.25).

Step-by-step explanation:

The inverse of the model c = 500 × 1.25^n can be found by interchanging the roles of c and n and then solving for n. To find the inverse, we will follow these steps:

1. Replace c with n and n with c to get n = 500 × 1.25^c.
2. Divide both sides by 500 to isolate the exponential term: 1.25^c = n/500.
3. Apply the logarithm to both sides to remove the exponent: log(1.25^c) = log(n/500).
4. Use the property of logarithms that log(a^b) = b × log(a) to simplify: c × log(1.25) = log(n/500).
5. Finally, divide both sides by log(1.25) to solve for c: c = log(n/500) / log(1.25).

This gives us the inverse function of the cost model, which tells us how many cards n we can produce for a certain cost c.