Question

The Ship It Anywhere Company bought a truck for $245,000. According to the company’s accounting department, the truck will depreciate $32,500 per year. 1. Find a linear function V(t) of the form V(t) = mt + b that models the value of the truck. V is the value of the truck and t is the number of years after the truck was bought. a. What is the slope of the function? Interpret what the slope means. b. What is the V intercept? Interpret what the V intercept means. c. Give the formula for the function. 2. Usethefunctiontofindthetintercept.Interpretwhatthetinterceptmeans. 3. Graphthefunction. 4. What is the domain and range of V(t)? 5. Find V(8) and explain what it means. Does your answer make sense? 6. When will the truck have a value of $128,000? 7. When will the truck have a value between $62,000 and $140,000?

Answer 1

1.
`V(t) = -32500t + 245000`

a. The slope of the function is m = -32500, and it means the change in the value of V(t) for each unitary change in the value of t.

b. The V-intercept is b = 245000, and it means the value of V(t) when t = 0, that is, the inicial value of V(t).

c. The formula is:
`V(t) = -32500t + 245000`

2. t-intercept:
`t = 7.5385`

The t-intercept means when the function V(t) will be zero, that is, the truck has no value anymore.

3. Graph in the image attached.

4. The domain is t = [0, 7.5385] and the range is V(t) = [245000, 0].

5.
`V(8) = -15000`

It means the price the truck will have after 8 years. It does not make sense, because the truck can't have a negative price.

6. After 3.6 years.

7. Between 3.23 years and 5.63 years.

Explanation:

1.

The inicial value is 245,000, and each year the value decreases 32,500, so we can write the equation:

`V(t) = -32500t + 245000`

a. The slope of the function is m = -32500, and it means the change in the value of V(t) for each unitary change in the value of t.

b. The V-intercept is b = 245000, and it means the value of V(t) when t = 0, that is, the inicial value of V(t).

c. The formula is:
`V(t) = -32500t + 245000`

2.

To find the t-intercept we just need to use V(t) = 0 and then find the value of t:

`0 = -32500t + 245000`

`32500t = 245000`

`t = 7.5385`

The t-intercept means when the function V(t) will be zero, that is, the truck has no value anymore.

3.

The graph of the function is in the image attached.

4.

The domain is t = [0, 7.5385] and the range is V(t) = [245000, 0].

5.

`V(8) = -32500*8 + 245000 = -15000`

It means the price the truck will have after 8 years. It does not make sense, because the truck can't have a negative price.

6.

`128000 = -32500t + 245000`

`32500t = 117000`

`t = 3.6`

After 3.6 years.

7.

`62000 = -32500t + 245000`

`32500t = 183000`

`t = 5.6308`

`140000 = -32500t + 245000`

`32500t = 105000`

`t = 3.2308`

Between 3.23 years and 5.63 years.