Sam bought a new car for $26,500. The car depreciates approximately 13% of its value each year.
A) What is the decay factor for the value of this car?
b) Write an equation to model the decay value of this car?
c) What will the value of the car be in 10 years?
a. DF =(1-r)^t
DF = Decay Factor.
r = Depreciation rate expressed as a
decimal.
t = Time in years.
b. V = Vo*DF = Vo*(1-r)^t.
V = Value after t years.
Vo = Initial Value or value when new.
c. V = 26,500(1-0.13)^10 = $6,583.22
A) The decay factor for the value of this car can be calculated by subtracting the depreciation percentage from 100% and converting it to a decimal.
Decay factor = (100% - 13%) / 100% = 87% / 100% = 0.87
b) To model the decay value of this car, we can use the equation:
Value after t years = Initial value x Decay factor^t
Let V represent the value of the car after t years, and P represent the initial purchase price of the car. The equation can be written as:
V(t) = P x (0.87)^t
c) To find the value of the car in 10 years, we substitute t = 10 into the equation:
V(10) = $26,500 x (0.87)^10
V(10) ≈ $9,734.98
Therefore, the value of the car in 10 years will be approximately $9,734.98.
To answer these questions, we need to understand how decay works in this context. Decay refers to the gradual decrease in value over time. In this case, the car depreciates by approximately 13% each year.
a) To find the decay factor, we can use the formula:
decay factor = (100% - percentage decrease) / 100%
In this case, the percentage decrease is 13%. Substituting this value into the formula:
decay factor = (100% - 13%) / 100%
decay factor = 87% / 100%
decay factor = 0.87
Therefore, the decay factor for the value of this car is 0.87.
b) To write an equation to model the decay value of this car, we can use the formula:
value after decay = initial value * decay factor^time
In this case, the initial value is $26,500, and the decay factor is 0.87. The "time" in this equation represents the number of years.
So the equation is:
value after decay = $26,500 * 0.87^time
c) To calculate the value of the car in 10 years, we can substitute the value of "time" with 10 in the equation:
value after decay = $26,500 * 0.87^10
Using a calculator, we can evaluate this expression to find the value of the car in 10 years.