Question Statement

If the inflation rate is continuously compounded $4\%$ per year and the price of a commodity is \50$ today:

a. Derive the function for the price of the commodity over time.

b. Find the price after 8 years.

c. Find the instantaneous rate of price at $t=8$ years.

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Background and Explanation

This problem applies the continuous compounding model, where quantities grow exponentially at a rate proportional to their current value. You will need the continuous growth formula and basic differentiation rules for exponential functions to solve for future values and instantaneous rates of change.

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Solution

Part (a)

The general formula for continuous compounding (exponential growth) is:

$P(t)=P_0{e}^{rt}$

where:

• $P_0$ is the initial price
• $r$ is the continuous growth rate (as a decimal)
• $t$ is time in years

Given the initial price $P_0=50$ and the inflation rate $r=0.04$ (which is $4\%$), we substitute these values into the formula:

$P(t)=50e^{(0.04)t}$

or simply:

$P(t)=50e^{0.04t}$

Part (b)

To find the price after 8 years, substitute $t=8$ into the function derived in part (a):

$\begin{array}{rl}P(8)& =50e^{(0.04)(8)}\\ & =50e^{0.32}\\ & =50(1.37713)\\ & =68.85\end{array}$

Therefore, the price after 8 years is approximately \68.85$.

Part (c)

The instantaneous rate of price change is given by the derivative \frac{dP}{dt}.

Starting with P(t) = 50e^{0.04t}, we differentiate with respect to $t$ using the chain rule (the derivative of $e^{kt}$ is $k{e}^{kt}$):

$\frac{dP}{dt}=50\cdot (0.04)\cdot e^{0.04t}=2e^{0.04t}$

Now evaluate this at $t=8$:

$\begin{array}{rl}\frac{dP}{dt}& =50(0.04)e^{(0.04)(8)}\\ & =(50)(0.04)e^{0.32}\\ & =(50)(0.04)(1.37713)\\ & =2.75\end{array}$

Thus, the instantaneous rate of price increase at $t=8$ years is approximately \2.75$ per year.

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Key Formulas or Methods Used

• Continuous compounding formula: $P(t)=P_0{e}^{rt}$
• Differentiation of exponential functions: $\frac{d}{dt}[e^{kt}]=k{e}^{kt}$ (applied here as $\frac{dP}{dt}=r{P}_0{e}^{rt}$)
• Chain rule for differentiating composite exponential functions

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Summary of Steps

1. Identify parameters: Recognize $P_0=50$ and $r=0.04$ for the continuous growth model
2. Establish the price function: Write $P(t)=50e^{0.04t}$
3. Calculate future price: Substitute $t=8$ to find $P(8)=50e^{0.32}\approx 68.85$
4. Find the rate function: Differentiate to get $\frac{dP}{dt}=2e^{0.04t}$ (or $50(0.04)e^{0.04t}$)
5. Evaluate instantaneous rate: Substitute $t=8$ into the derivative to obtain $2.75$ dollars per year