A small business owner buys a truck for $20,000 to transport supplies for her business. She anticipates that she will use the truck for 5 years and that the truck will be worth $10,000 in 5 years. She plans to claim a depreciation tax credit using the straight-line depreciation method approved by the Internal Revenue Service. This means that if V is the value of the truck at time t, then a linear equation is used to relate V and t.

Question

A small business owner buys a truck for $20,000 to transport supplies for her business. She anticipates that she will use the truck for 5 years and that the truck will be worth $10,000 in 5 years. She plans to claim a depreciation tax credit using the straight-line depreciation method approved by the Internal Revenue Service. This means that if V is the value of the truck at time t, then a linear equation is used to relate V and t.

(a) Find a linear equation that models the depreciated value V of the truck t years since it was purchased.

(b) What is the rate of depreciation?

I've tried but I'm really stuck.

Answer 1

All you want is the linear equation passing through the two points

(0,20000) and (5,10000)
slope = (10000-20000)/(5-0) = -2000

So if we let t be the time

V = -2000t + 20000

the rate of depreciation is $2000/year

Answer 2

To find a linear equation that models the depreciated value V of the truck t years since it was purchased, we can use the straight-line depreciation method.

The straight-line depreciation method assumes that the value of the truck decreases by a fixed amount each year. We can use the formula for slope-intercept form of a linear equation, y = mx + b, where y is the value of the truck at time t, x is the number of years since the purchase, m is the rate of depreciation, and b is the initial value of the truck.

(a) To find the linear equation, we need to determine the rate of depreciation and the initial value of the truck.

First, we know that the truck was purchased for $20,000 (the initial value) and it is expected to have a value of $10,000 in 5 years.

Using the formula for slope (m) in the slope-intercept form, we can calculate the rate of depreciation:
m = (change in value of the truck) / (change in time)
m = (value at time t - initial value) / (time)

m = ($10,000 - $20,000) / (5 years - 0 years)
m = -$10,000 / 5 years
m = -$2,000

So, the rate of depreciation is -$2,000 per year (negative because the value is decreasing).

Now, we can substitute the values of m (rate of depreciation) and b (initial value) into the equation y = mx + b.

Using y for V and x for t, the linear equation is:
V = -2000t + 20000

(b) The rate of depreciation is -$2,000 per year. This means that the value of the truck decreases by $2,000 for each year that passes.