Jimmy bought a car in January of 2010 worth $25, 500. He submitted the car for valuation in January 2015 and was told that its worth is now $20, 450. Assuming that the depreciation is linear.

a) Find the rate at which the car is decreasing in value. Treat the year 2000 as 0 t  .
b) Interpret the value obtained in part (a).
c) Complete the equation of the form V mt c  .
d) Use your equation in part (c) to estimate the value of the car in 2019.
e) Jimmy wants to sell the car when its value is $10,000. In what year will he most likely sell this car?

a) To find the rate at which the car is decreasing in value, we can use the formula for slope (rate of change) which is given by:

Rate of change = (change in value) / (change in time)

The change in value is $25,500 - $20,450 = $5,050
The change in time is 2015 - 2010 = 5 years

Rate of change = $5,050 / 5 = $1,010 per year

b) The value obtained in part (a) represents the average rate at which the car is decreasing in value per year. This means that every year, the car's value is decreasing by $1,010.

c) The equation of the form V = mt + c represents the linear relationship between the value of the car (V) and time (t). To complete the equation, we need to find the values of m and c.

We know that in January 2010 (t = 0), the value of the car is $25,500. Plugging this into the equation, we get:

$25,500 = m(0) + c
$25,500 = c

Therefore, the equation becomes V = mt + $25,500.

d) To estimate the value of the car in 2019, we can plug in t = 2019 into the equation:

V = m(2019) + $25,500

We need to find the value of m in order to complete the equation. To do this, we can use the rate of change we calculated in part (a):

$1,010 = m

Plugging this into the equation, we get:

V = $1,010(2019) + $25,500

Simplifying this gives us the estimated value of the car in 2019.

e) To find the year Jimmy will most likely sell the car when its value is $10,000, we need to solve the equation when V = $10,000:

$10,000 = $1,010t + $25,500

Rearranging the equation:

$1,010t = $10,000 - $25,500
$1,010t = -$15,500

Dividing both sides by $1,010:

t = -15,500 / 1,010

Since we assumed the year 2000 as t = 0, the year Jimmy will most likely sell the car is 2000 + (-15,500 / 1,010).

a) To find the rate at which the car is decreasing in value, we can calculate the depreciation rate. The depreciation rate is the change in value divided by the change in time.

First, let's determine the change in value:

Change in value = Final value - Initial value
Change in value = $20,450 - $25,500
Change in value = -$5,050 (Note: Negative value represents a decrease)

Next, let's determine the change in time:

Change in time = Final year - Initial year
Change in time = 2015 - 2010
Change in time = 5 years

Now, we can calculate the depreciation rate:

Depreciation rate = Change in value / Change in time
Depreciation rate = -$5,050 / 5 years
Depreciation rate = -$1,010 per year

Therefore, the car is decreasing in value at a rate of $1,010 per year.

b) The value obtained in part (a) represents the average annual rate at which the car is depreciating. It means that, on average, the value of the car is decreasing by $1,010 each year.

c) To complete the equation in the form V = mt + c, we need to determine the initial value (V0) and the constant (c).

We know that the initial value of the car (V0) in 2010 was $25,500. Using this information, we can set up the equation as follows:

V = mt + c
$25,500 = 2010m + c

We do not have enough information to determine the constant (c) in this scenario.

d) To estimate the value of the car in 2019 using the linear depreciation model, we can use the equation from part (c):

V = mt + c
$V2019 = 2019m + c

Since we don't know the constant (c), we can't provide an exact value for the car in 2019. However, we can estimate it by substituting the average depreciation rate we calculated in part (a).

$V2019 = 2019 * (-$1,010) + c
$V2019 = -$2,034,090 + c

e) To determine in which year Jimmy will most likely sell the car when its value is $10,000, we can use the equation from part (c).

V = mt + c
$10,000 = mt + c

We need to solve this equation for the time (t) variable. However, since we do not have the constant (c), we cannot provide an exact value for the year.