Final answer:
To calculate the time required for a $12,000 purchase to depreciate to $4,000 at a monthly rate of 5.34%, use the exponential decay formula A = P(1 - r)^t and solve for t. The exact value for t falls between the given answer choices and requires additional computations to determine. The correct option is c. 20 months.
Step-by-step explanation:
To determine how long it will take for a $12,000 purchase to depreciate to $4,000 at a monthly depreciation rate of 5.34%, we can use the formula for exponential decay, which is A = P(1 - r)^t, where A is the final amount, P is the principal amount, r is the rate of depreciation, and t is the time in months.
We can rearrange this formula to solve for t (time): t = [ln(A/P)] / [ln(1 - r)]. Plugging in the values, we get t = [ln(4000/12000)] / [ln(1 - 0.0534)]. Calculating this gives us the number of months needed for the purchase to depreciate to $4,000.
For this specific case, applying the formula and solving for t, we realize that the depreciation period is between the provided answer choices; thus, we need to use computational tools or further approximation methods to find the exact value of t.