Question

A $16,000 robot depreciates linearly to zero in 10 years. (a) find a formula for its value as a function of time, t, in years.

Answer 1

We are given, cost of the robot for 0 number of year = $16,000.

0 represents initial time of the robot.

After 10 year cost of the robot is = $0

The problem is about the number of the years and cost of the robot over different number of years.

So, we could take x coordinate by number of hours and y coordinate for y number of hours.

So, from the problem, we could make two coordinates for the given situation.

(x1,y1) = (0, 16000) and (x2,y2) = (10, 0).

In order to find the function of time, we need to find the rate at which robot rate depreciates each year.

Slope is the rate of change.

So, we need to find the slope of the two coordinates we wrote above.

We know, slope formula

$Slope (m) = \frac{y2-y1}{x2-x1}$

Plugging values in formula, we get

$m=\frac{0-16000}{10-0} = \frac{-16000}{-10} = -1600.$

Brecasue of depreciation we got a negative number for slope or rate of change.

Therefore, rate of depreciation is $1600 per year.

We already given inital cost, that is $16,000.

So, we can setup an a function

f(x) = -1600x + 16000.

But the problem is asked to take the variable t for time.

Replacing x by t, we get

f(t) = -1600t + 16000.