you purchase an suv for $26,000. A year later the car is worth only $24,800. If the value of the car continues to depreciate at that rate,
a. find the linear equation that determines the value of the car based on the number of years you own it.
b. when will the car be worth $500?
rate = -1200 per year
Value = -1200t + 26000
b) set value = 500
500 = -1200n + 26000
1200n = 21000
n = 17.5
In about 17 and a half years
To find the linear equation that determines the value of the car based on the number of years you own it, we need to know the equation of a line, which is of the form y = mx + b, where y represents the value of the car, x represents the number of years you own it, m represents the slope or rate of depreciation, and b represents the initial value of the car.
a. To determine the slope (m) of the line, we need to find the rate at which the car's value decreases per year. The rate of depreciation can be calculated as the difference in value over the difference in time. In this case, the car's value decreases by $200 over a period of 1 year, thus the rate of depreciation (m) is -200.
Next, we need to find the initial value (b) of the car. Given that the car is worth $26,000 initially (when x = 0), we know that b = 26,000.
Now we can write the linear equation using the slope and the initial value:
y = mx + b
y = -200x + 26,000
So, the linear equation that determines the value of the car based on the number of years you own it is y = -200x + 26,000.
b. To find when the car will be worth $500, we need to solve the equation y = -200x + 26,000 for x.
Substituting y = 500 into the equation:
500 = -200x + 26,000
Next, we can isolate x by moving -200x to the other side:
-200x = 500 - 26,000
-200x = -25,500
To solve for x, we divide both sides of the equation by -200:
x = (-25,500) / (-200)
x = 127.5
Therefore, the car will be worth $500 after approximately 127.5 years of ownership.