Problem 5 - Calculus Early Transcendentals

Question

How are the rate constant and the doubling time related?

Step-by-Step Solution

Answer

Answer: The rate constant (k) and the doubling time (T_d) in an exponential growth model are related by the formula k = ln(2)/T_d. As the rate constant increases, the doubling time decreases, indicating a faster growth, while a decrease in the rate constant leads to an increase in the doubling time, signifying slower growth.

Step 1: Write the exponential growth equation

The exponential growth equation describing the growth of a quantity with time is given by: N(t) = N_0 * e^{k*t} where N(t) is the amount of the quantity at time t, N_0 is the initial amount, k is the rate constant, and t is time.

Step 2: Define the doubling time

The doubling time, denoted as T_d, is the time taken for the quantity to double. So, at time t = T_d, N(t) = 2 * N_0.

Step 3: Substitute the doubling time into the exponential growth equation

Now, substitute the doubling time T_d in the equation from Step 1: 2 * N_0 = N_0 * e^{k*T_d}

Step 4: Solve for the rate constant k

Divide both sides of the equation by N_0: 2 = e^{k*T_d} Next, take the natural logarithm (ln) of both sides: ln(2) = k*T_d Finally, solve for the rate constant k: k = \frac{ln(2)}{T_d}

Step 5: Write the conclusion

The rate constant (k) and the doubling time (T_d) are related by the formula: k = \frac{ln(2)}{T_d} This formula shows that as the rate constant increases, the doubling time decreases, meaning that the growth is faster. Conversely, as the rate constant decreases, the doubling time increases, indicating slower growth.