When the annual rate of inflation averages 6% over the next 10 years, the approximate cost C of goods or services during any year in that decade is given below, where t is the time in years and P is the present cost.
C(t) = P(1.06)t
(a) The price of an oil change for your car is presently $24.40. Estimate the price 10 years from now. (Round your answer to two decimal places.)
C(10) = $ Correct: Your answer is correct.


(b) Find the rates of change of C with respect to t when
t = 1
and
t = 5.
(Round your coefficients to three decimal places.)
At
t = 1
	
Incorrect: Your answer is incorrect.
At
t = 5
  	
Incorrect: Your answer is incorrect.


(c) Verify that the rate of change of C is proportional to C. What is the constant of proportionality?
Correct: Your answer is correct. 

Let's solve each part of the problem step by step.

### (a) Estimating the price 10 years from now
The formula provided is:

\[
C(t) = P(1.06)^t
\]

Given:
- $ P = 24.40 $ (present cost of the oil change)
- $ t = 10 $ years

We need to calculate $ C(10) $:

\[
C(10) = 24.40 \times (1.06)^{10}
\]

Let's compute this:

\[
C(10) = 24.40 \times 1.790847
\]

\[
C(10) \approx 43.69
\]

So, the estimated price 10 years from now is **$43.69**.

### (b) Finding the rates of change of $ C $ with respect to $ t $ at $ t = 1 $ and $ t = 5 $
The rate of change of $ C(t) $ with respect to $ t $ is the derivative of $ C(t) $ with respect to $ t $.

Given $ C(t) = P(1.06)^t $, let's find the derivative:

\[
\frac{dC}{dt} = P \cdot \frac{d}{dt}\left[(1.06)^t\right]
\]

Using the chain rule, where $ \frac{d}{dt}(a^t) = a^t \ln(a) $:

\[
\frac{dC}{dt} = P \cdot (1.06)^t \cdot \ln(1.06)
\]

Now let's evaluate this derivative at $ t = 1 $ and $ t = 5 $ with $ P = 24.40 $.

1. **At $ t = 1 $:**

\[
\frac{dC}{dt} \bigg|_{t=1} = 24.40 \cdot (1.06)^1 \cdot \ln(1.06)
\]

\[
\frac{dC}{dt} \bigg|_{t=1} \approx 24.40 \cdot 1.06 \cdot 0.05827 \approx 1.507
\]

2. **At $ t = 5 $:**

\[
\frac{dC}{dt} \bigg|_{t=5} = 24.40 \cdot (1.06)^5 \cdot \ln(1.06)
\]

\[
\frac{dC}{dt} \bigg|_{t=5} \approx 24.40 \cdot 1.3382 \cdot 0.05827 \approx 1.900
\]

### (c) Verifying the rate of change of $ C $ is proportional to $ C $
We want to show that $ \frac{dC}{dt} $ is proportional to $ C(t) $. 

From the derivative calculated earlier:

\[
\frac{dC}{dt} = P(1.06)^t \ln(1.06)
\]

We know $ C(t) = P(1.06)^t $, so:

\[
\frac{dC}{dt} = C(t) \cdot \ln(1.06)
\]

This shows that the rate of change $ \frac{dC}{dt} $ is proportional to $ C(t) $ with a constant of proportionality $ \ln(1.06) $, which is approximately **0.0583**.

Let me know if you'd like more details, or if you have any further questions! Here are 5 related questions:

1. How would the formula change if the inflation rate was compounded monthly instead of annually?
2. If the inflation rate were 5% instead of 6%, what would be the cost in 10 years?
3. Can you derive a general formula for the price after $ t $ years for any inflation rate $ r $?
4. How would a decrease in the inflation rate to 4% affect the rate of change of $ C(t) $?
5. What would be the cost after 10 years if the initial price were $30.00 instead of $24.40?

**Tip:** When solving for the rate of change, always check if the function itself might help simplify the derivative, as seen with exponential functions.

When the annual rate of inflation averages 6% over the next 10 years, the approximate cost C of goods or services during any year in that decade is given below, where t is the time in years and P is the present cost. C(t) = P(1.06)^t (a) The price of an oil change for your car is presently $24.40. Estimate the price 10 years from now. (Round your answer to two decimal places.) (b) Find the rates of change of C with respect to t when t = 1 and t = 5. (Round your coefficients to three decimal places.) (c) Verify that the rate of change of C is proportional to C. What is the constant of proportionality?

Learn how to calculate the future price of goods and services with exponential growth due to a 6% inflation rate. This problem includes the formula for exponential growth, rates of change, and the concept of proportionality. Suitable for students in Grades 10-12.

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