A new car with a purchased price of $23,500 has a value of $15,300 two years later.
a) Write the straight-line model: V= mt+b
b) Write the exponential model: V= ae^(kt)
c) Interpret the meaning of the slope in the straight-line model.
a)
The fact that the model is LINEAR tells you that the depreciation is a constant amount every year.
So if after two years the car is worth 15,300, that means that over the two years we have lost 23,500 - 15,300 = 8,200, which is 4100 per year.
So 4100 is the amount you lose each year.
So at any given year t, the value V of your car will have depreciated by 4100*t.
So your model is V(t) (the value after t years) = -4100*t (the yearly depreciation by the number of years) + 23,500 (the initial value).
b)
For the exponential model, we know it satisfies
at time 0
23,500 = 23,500e^(k*0) ... which isn't much use, but important to understand,
and more usefully at time 2
15,300 = 23,500e^(k*2)
So we need to solve for k:
taking the natural log of both sides:
ln (15,300/23,500) = 2k
-0.429147593 = 2k
k = -0.214573796
So your exponential model is given by
V=23,500e^(-0.215* t )
c) The answer should be more or less given in part a there...
a) Substitute t = 0 and V = 23,500: 23,500 = b. Therefore V = mt + 23,500.
Now substitute t = 2 and V = 15,300 and solve for m.
b) Substitute t = 0 and V = 23,500: 23,500 = a. Therefore V = 23,500 e^(kt).
Now substitute t = 2 and V = 15,300 and solve for k (the previous poster has explained how to do this).
c) Depreciation per year.
If you need more help, you will need to say what it is that you don't understand.
a) Your model is V(t) (the value after t years) = -4100*t (the yearly depreciation by the number of years) + 23,500 (the initial value).
b)
V=23,500e^(-0.215* t )
For d)
Your formula from a is V(t) = -4100t + 23,500
Where t is the number of years, and V(t) is the value after t years.
So if you are asked to find the value after 5 years, you simply fill in 5 instead of t, giving
V(5) = -4100*5 + 23,500
V(5) = 3000, so the value after 5 years is 3000.
The answer for the exponential model is got in a similar fashion - just fill in the number of years instead of t and you'll get an answer for V(t).