Suppose a car that sells for $40,000 depreciates 10% per year. How many years would it take for the car to have a value less than $25,000

Question

Suppose a car that sells for $40,000 depreciates 10% per year. How many years would it take for the car to have a value less than $25,000

Answer 1

40000(.9)^n < 25000

.9^n < .625

take log of both sides and use log rules

n log .9 < log .625
n < 4.46 yrs

testing:
40000(.8)^4.5 = 24,897.24
40000(.9)^4 = 26244
4000(.9)^5 = 23619.60

So, I would say in 4 1/2 years

Answer 2

To determine the number of years it would take for the car to have a value less than $25,000, we can use the concept of exponential decay. The general formula for exponential decay is:

V = V₀ * (1 - r)^t

where:
V is the final value
V₀ is the initial value
r is the decay rate
t is the number of years

In this case, the initial value (V₀) is $40,000, the decay rate (r) is 10% or 0.10, and the final value (V) is $25,000. We need to solve for the number of years (t).

$25,000 = $40,000 * (1 - 0.10)^t

To solve for t, we need to isolate it on one side of the equation. Let's divide both sides by $40,000:

($25,000 / $40,000) = (1 - 0.10)^t

0.625 = (0.9)^t

To solve for t, we can take the natural logarithm (ln) of both sides:

ln(0.625) = ln(0.9)^t

Using logarithm rules, we can bring down the exponent:

ln(0.625) = t * ln(0.9)

Now, divide both sides by ln(0.9):

t = ln(0.625) / ln(0.9)

Using a calculator, we can compute the value of t:

t ≈ 4.94

Therefore, it would take approximately 4.94 years for the car to have a value less than $25,000.

Answer 3

To find out how many years it would take for the car to have a value less than $25,000, we need to determine the depreciation per year. Given that the car depreciates 10% per year, the value after each year can be calculated by subtracting 10% of the previous year's value.

Let's break down the calculation step by step:

1. Start with the initial value of the car, which is $40,000.
2. Calculate the depreciation for the first year: $40,000 * 0.10 = $4,000.
3. Subtract the depreciation from the initial value to get the value after the first year: $40,000 - $4,000 = $36,000.
4. Now repeat the process for subsequent years. For the second year, the depreciation is calculated based on the value after the first year: $36,000 * 0.10 = $3,600. Subtract this depreciation from the value after the first year to get the value after the second year.
5. Continue this process until the value falls below $25,000.

Now, to determine the number of years it takes for the car's value to be less than $25,000, we can use a loop or a mathematical formula based on the depreciation rate.

Let's use the loop approach, assuming the car's value is reduced after each year:

```
initial_value = 40000
target_value = 25000
depreciation_rate = 0.10

years = 0

while initial_value > target_value:
depreciation = initial_value * depreciation_rate
initial_value -= depreciation
years += 1

print(years)
```

By running this code, we find that it would take approximately 4 years for the car to have a value less than $25,000, assuming it depreciates 10% per year.

Answer 4

Make the following model:

40000-400x < 25000

Plug in some numbers for x.