Question

A function models a linear relationship between the mileage (miles driven) for a truck and the number of years since it was purchased. Currently, the truck’s mileage is 107,500, and it has been driven at a rate of 15,000 miles per year since it was purchased 7 years ago.

What is the initial value of the mileage function, in miles?
a. 0 miles
b. 7 miles
c. 2,500 miles
d. 10,700 miles

Answer 1

To determine the initial value of the mileage function, we can use the information given:

- The truck has been driven at a rate of 15,000 miles per year.
- It has been driven for 7 years.

The mileage after 7 years is given as 107,500 miles. We can express the mileage as a function of the number of years since it was purchased (let's denote \( t \) for years since purchased). The relationship can be modeled by the equation:

\[
\text{Mileage} = \text{Initial Mileage} + \text{Rate of Mileage Increase} \times t
\]

We can represent this as:

\[
M(t) = M_0 + 15,000t
\]

Where \( M_0 \) is the initial mileage (the mileage when the truck was first purchased), and \( t \) is the number of years since it was purchased.

After 7 years, we know:

\[
M(7) = 107,500
\]

Substituting \( t = 7 \) into the equation gives us:

\[
107,500 = M_0 + 15,000 \times 7
\]
\[
107,500 = M_0 + 105,000
\]

To find \( M_0 \), we can rearrange the equation:

\[
M_0 = 107,500 - 105,000
\]
\[
M_0 = 2,500
\]

So, the initial value of the mileage function, representing the mileage when the truck was purchased, is:

**c. 2,500 miles**