Question

15–20. Designing exponential growth functions Complete the following steps for the given situation.

a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.
b. Answer the accompanying question.

Rising costs Between 2010 and 2016, the average rate of inflation was about 1.6%/yr. If a cart of groceries cost \$100 in 2010, what will it cost in 2025, assuming the rate of inflation remains constant at 1.6%?

Accepted Answer

Identify the general form of the exponential growth function, which is given by $$P(t) = P_0$ e$$^{k t}$$, where P_0$ is the initial amount, k$ is the growth rate constant, and t$ is the time in years. Assign the known values: the initial price P_0$ = 100$$ dollars at time $$t = 0 ($$year 2010), and the price at $$t = 6$$ years (year 2016) reflects a 1.6% annual inflation rate. Since the inflation rate is 1.6%, the price grows by 1.6% each year, so $$P(6) = 100 	imes (1 + 0.016)^6$. Use the exponential growth formula to set up the equation for P(6$$)$$: 100$$ $$e^{6k} = 100 	$$imes $$(1.016)^6. $$Simplify this to find $$e^{6k}$$ = $$(1.016)^6. $$Take the natural logarithm of both sides to solve for $$k$$: $$6k = \ln((1.016)^6$)$$, which simplifies to $$6k = 6 \ln(1.016). $$Then, solve for $$k by $$dividing both sides by 6, giving $$k = \ln(1.016). $$With $$k$$ found, write the exponential growth function as $$P(t) = 100 e$$^{k t}$$. To find the cost in 2025, calculate P(15$$)$$ since 2025 is 15 years after 2010, by substituting t = 15$ into the function.

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Key Concepts

Exponential Growth Function

Rate Constant (k) in Exponential Growth

Applying Exponential Growth to Inflation