Question

Mr. Passwater decides to buy a brand new 2024 MINI Cooper convertible for $39,645. The value of his car is expected to depreciate (decrease) by 17% each year for the first five years. Let ◤ represent the value of Mr. Passwater’s MIN1 Cooper at time t years after purchase.

Let V(m)=ak ''' be an equivalent expression for V(m) where m is the number of months after purchase. Find the value of k.

Answer 1

To find the value of 'k' in the expression V(m) = ak, we need to relate the number of months 'm' to the number of years 't'. By using the conversion factor of 1 year = 12 months, we can substitute t = m/12 into the depreciation equation to find the value of k.

Step-by-step explanation:

To find the value of k in the expression V(m) = ak, we need to relate the number of months m to the number of years t. Since there are 12 months in a year, we can use the conversion factor of 1 year = 12 months. This means that t = m/12 (dividing m by 12).

The value of the car after t years can be expressed as:

◤ = 39,645 * (1 - 0.17)^t

Substituting t = m/12 into the equation gives:

◤ = 39,645 * (1 - 0.17)^(m/12)

Comparing this to the expression V(m) = ak, we can see that k = (1 - 0.17)^(1/12), which simplifies to 0.916 (rounded to three decimal places).

Answer 2

To find the monthly depreciation constant k for the car's value, we convert the annual rate of 17% to a monthly rate by taking the twelfth root of the remaining value (0.83), resulting in k ≈ 0.985.

Step-by-step explanation:

The student's question is about determining the constant k in the exponential decay formula for the value of a car over time. Mr. Passwater purchases a car for $39,645 and it depreciates by 17% annually. To find the value of k, we need to convert the annual depreciation into monthly depreciation since m represents the number of months.

Firstly, if the value depreciates by 17% annually, it means that the car retains 83% of its value each year (100% - 17%). To find the monthly factor, we take the twelfth root of 0.83 (since there are 12 months in a year). Mathematically, this is expressed as k = 0.83^(1/12).

Substituting the values in, we get k = 0.83^(1/12) which gives us k ≈ 0.985 when calculated.