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.

Cell growth The number of cells in a tumor doubles every 6 weeks starting with 8 cells. After how many weeks does the tumor have 1500 cells?

Accepted Answer

Step 1: Identify the general form of the exponential growth function, which is given by $$N(t) = N_0$ e$$^{k t}$$, where N(t$$)$$ is the number of cells at time t$, N_0$ is the initial number of cells, k$ is the growth rate constant, and t$ is the time in weeks. Step 2: Use the information that the number of cells doubles every 6 weeks to find the rate constant k$. Since doubling means N(6) = 2$$ $$N_0$$, substitute into the formula: $$2 N_0$ = N_0$ e$$^{k 	imes 6}$$. Simplify this to 2$$ = $$e^{6k}. $$Step 3: Solve for $$k by $$taking the natural logarithm of both sides: $$\ln(2) = 6k$$, which gives $$k = \frac{\ln(2)}{6}. $$Step 4: Write the exponential growth function using the initial number of cells $$N_0 = 8$ and the rate constant k$: N(t) = 8$$ $$e^{\frac{\ln(2)}$${6} t}$$. Step 5: To find the time t$ when the tumor has 1500 cells, set N(t) = 1500$ and solve for t$: 1500 = 8$$ $$e^{\frac{\ln(2)}$${6} t}$$. Divide both sides by 8, then take the natural logarithm to isolate t$: $$\ln\left(\frac{1500}{8}ight) = \frac{\ln(2)}{6} t$$, and finally solve for t$ by multiplying both sides by $$\frac{6}{\ln(2)}$$.

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Key Concepts

Exponential Growth Function

Rate Constant (k) in Exponential Growth

Solving for Time in Exponential Equations