A $40 000 car depreciates at a rate of 13% per year.
a) Write an equation to represent this situation.
b) Find the value of the car in 3 years.
c) How long does it take for the car to be worth half it's value?
Thanks
a) To write an equation representing this situation, we can use the formula for exponential decay:
V = P(1 - r)^t
Where:
V = value of the car at a given time
P = initial value of the car
r = rate of depreciation (as a decimal)
t = time in years
In this case, the initial value of the car is $40,000, and the rate of depreciation is 13% per year, or 0.13 as a decimal. Therefore, the equation for this situation is:
V = 40000(1 - 0.13)^t
b) To find the value of the car in 3 years, we can substitute t = 3 into the equation we found in part (a):
V = 40000(1 - 0.13)^3
Now, we can solve this equation using a calculator or by simplifying the expression in the parentheses:
V = 40000(0.87)^3
V ≈ 40000(0.658503)
V ≈ 26340.12
Therefore, the value of the car after 3 years is approximately $26,340.12.
c) To find how long it takes for the car to be worth half its value, we need to solve the equation V = 0.5P using the formula we derived in part (a):
0.5P = P(1 - 0.13)^t
Dividing both sides by P and simplifying, we get:
0.5 = (1 - 0.13)^t
Now, we can solve this equation for t by taking the logarithm of both sides with base (1 - 0.13):
log(base 0.87)(0.5) = t
Using a calculator, we find:
t ≈ 3.92
Therefore, it takes approximately 3.92 years for the car to be worth half its value.
future value=present value*(1-i)^time
present value=40000
i=.13
equation:
future value=40,000(.87)^t
value in three years: Put this in your google search window:
40000(.87)^3=
how long? take the log of each side:
log20,000=log40,000 + tLog.87
log20,000-log40,000=tlog.87
log(20/40=log(.5)=tlog.87
time=log(.5)/log.87
put this in your google search engine
log(.5)/log(.87) = and is should be rounded to the next year.