The formula for the value of a car, C, after t years of depreciation of its initial value v at a rate of r is C=v(1−r)t.
A: How can you rewrite the formula to correctly isolate v?
B: What was the initial value of the car (to the nearest dollar) if it depreciated 10% each year for 10 consecutive years and has a current value of $8,019.60?
HUH ?
each year new value = (1-r) * old value
so
final value C = v (1-r)^t
so
v = C / (1-r)^t
===================
v = 8019.60 / 0.90^10 = 8019.60 / 0.348678 = 23,000
A: To isolate v in the formula C = v(1 - r)^t, we can rewrite it as:
v = C / (1 - r)^t
B: To find the initial value of the car, we can substitute the given values into the formula:
C = $8,019.60
r = 10% = 0.10
t = 10 years
Plugging these values into the formula, we get:
v = $8,019.60 / (1 - 0.10)^10
Calculating the value inside the parentheses first:
(1 - 0.10)^10 = 0.9^10 ≈ 0.3487
Now substitute this value back into the formula:
v ≈ $8,019.60 / 0.3487
v ≈ $22,983.63 (rounded to the nearest dollar)
Therefore, the initial value of the car was approximately $22,984.
A: To isolate the initial value, v, in the formula C=v(1−r)t, you can rearrange the equation by dividing both sides by (1 - r)^t. This will give you:
v = C / (1 - r)^t
B: To find the initial value of the car, we can substitute the given information into the formula. We know that the current value, C, is $8,019.60. The rate of depreciation, r, is 10% or 0.10. And the number of years, t, is 10.
Using the formula v = C / (1 - r)^t, we can calculate the initial value:
v = $8,019.60 / (1 - 0.10)^10
First, subtract the depreciation rate from 1: (1 - 0.10) = 0.90
Next, raise that result to the power of the number of years: 0.90^10 ≈ 0.3487
Finally, divide the current value by the result above to get the initial value:
v ≈ $8,019.60 / 0.3487 ≈ $22,993.47
Therefore, to the nearest dollar, the initial value of the car was $22,993.