A car was bought in  
February 1, 2015
  for  
$
25
,
000
 , and its value depreciates by  
1.5
%
  every single month. The value of the car  
m
  months since  
February 1, 2015
  is given by the formula  
V
(
m
)
=
25
,
000
(
d
)
m
 , where  
d
  is the decay factor. Find the value of the car on  
June 1, 2017
.
  Round your answer to the nearest cent.



To find the value of the car on **June 1, 2017**, we'll break down the problem into steps:

1. **Find the number of months between February 1, 2015, and June 1, 2017:**

   - From **February 1, 2015**, to **February 1, 2017**, there are exactly 24 months.
   - From **February 1, 2017**, to **June 1, 2017**, there are 4 more months.

   Therefore, the total number of months, $ m $, is:
   \[
   m = 24 + 4 = 28 \text{ months}
   \]

2. **Determine the decay factor $ d $:**
   The car depreciates by **1.5%** every month, which means it retains **98.5%** of its value each month. So the decay factor $ d $ is:
   \[
   d = 1 - 0.015 = 0.985
   \]

3. **Use the given formula to find the car's value after $ m = 28 $ months:**

   The formula for the value of the car after $ m $ months is:
   \[
   V(m) = 25,000 \times (d)^m
   \]
   Substituting the values for $ d $ and $ m $:
   \[
   V(28) = 25,000 \times (0.985)^{28}
   \]

Now, let's calculate the value.

The value of the car on **June 1, 2017**, after 28 months of depreciation, is approximately **$16,373.99**, rounded to the nearest cent.

Would you like further details or explanations? Here are some related questions for deeper understanding:

1. What would be the car's value after 3 years of depreciation?
2. How does the car's value change if the depreciation rate is increased to 2% per month?
3. How many months would it take for the car's value to drop below $10,000?
4. How can we model depreciation with more complex factors, such as varying monthly rates?
5. What is the difference between exponential and linear depreciation?

**Tip:** In problems with percentage decay or growth, using the decay or growth factor simplifies calculations significantly.

A car was bought on February 1, 2015, for $25,000, and its value depreciates by 1.5% every month. The value of the car m months since February 1, 2015, is given by the formula V(m) = 25,000(d)^m, where d is the decay factor. Find the value of the car on June 1, 2017. Round your answer to the nearest cent.

Find the value of a car bought on February 1, 2015, for $25,000, depreciating by 1.5% every month, using exponential decay. This step-by-step guide covers the decay formula and calculates the car's value on June 1, 2017. Suitable for high school students (Grades 9-12).

Math Problem Statement

A car was bought in
February 1, 2015 for
$ 25 , 000 , and its value depreciates by
1.5 % every single month. The value of the car
m months since
February 1, 2015 is given by the formula
V ( m )

Solution

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