The Question is:
The value of a new car depreciates at a rate of 12% per year.
1)Write an equation to represent the approximate value of a car purchased for $23 000.
2)Determine the value of the car two years after it is purchased.
3)Approximately how many years will it take until the car is worth $2300?
I have received answers that differ for this question. I request feedback for further understanding what must be done.
The following is an example of how the answers I've received are different:
ex.1
a) V= C - rtC,
Eq: V = C(1 - rt).
V = value.
C = cost.
r = rate expresed as a decimal.
t = time in years.
b) V = 23000(1 - 0.12*2),
V = 23000 * 0.76 = 17480.
c) V = 23000(1 - 0.12t) = 2300,
23000(1 - 0.12t) = 2300,
Divide both sides by 23000:
1 - 0.12t = 0.1,
-0.12t = 0.1 - 1 = -0.90,
t = -0.90 / -0.12 = 7.5 yrs.
OR
a) A=23000(1-0.12)^1
A= 20240
b) A=23000(1-0.12)^2
A=17 811.20
c) 2300 = 23000(.88)^n
0.1 = (.88)^n
n = (log 0.1) / (log 0.88)
=18.01 years
No need for confusion:
This is an exponential function
value= 23000(.88)^t , where t is the number of years
2. when t=2
value = 23000(.88)^2 = $17811.20
3.
2300 = 23000(.88)^t
.1 = .88^t
take log of both sides
log .1 = log .88^t
log .1 = t log .88
t = log .1/log .88 = 18.01 or appr 18 years
Great this helps alot, was worried I screwed up something
To solve this problem, we can use the formula for exponential decay, which represents the value of an object as it depreciates over time.
The formula is given as:
V = C * (1 - r)^t
Where:
V is the value of the object after time t,
C is the initial value or cost of the object,
r is the rate of depreciation expressed as a decimal, and
t is the time in years.
Let's now solve the given questions using this formula.
1) Write an equation to represent the approximate value of a car purchased for $23,000.
Using the formula, the equation would be:
V = 23,000 * (1 - 0.12)^t
2) Determine the value of the car two years after it is purchased.
Substitute t = 2 into the equation:
V = 23,000 * (1 - 0.12)^2
V = 23,000 * 0.76
V = 17,480
Therefore, the value of the car two years after it is purchased is $17,480.
3) Approximately how many years will it take until the car is worth $2,300?
Set V = 2,300 in the equation and solve for t:
2,300 = 23,000 * (1 - 0.12)^t
Divide both sides by 23,000:
(1 - 0.12)^t = 2,300 / 23,000
Simplify:
(0.88)^t = 0.1
To find the value of t, we can take the logarithm of both sides.
t * log(0.88) = log(0.1)
t = log(0.1) / log(0.88)
t ≈ 18.01 years
Therefore, it will take approximately 18.01 years until the car is worth $2,300.
Both sets of answers provided are correct and may vary slightly based on rounding.