Chapter 24: Problem 19

Wheat growing is an example of

Short Answer

Expert verified

It will take approximately 4.32 years for the acreage of wheat to increase from 20,000 acres to 25,000 acres, given a growth rate of 5% per year.

Step by step solution

Identify the exponential growth formula

The exponential growth formula is given by: \(A = A_0 \cdot e^{rt}\), where A is the final amount, \(A_0\) is the initial amount, r is the growth rate, and t is time.

Plug in the given information

We are given the initial amount (\(A_0\)) as 20,000 acres, final amount (A) as 25,000 acres, and the growth rate (r) as 5% or 0.05 in decimal form. Substitute these values into the exponential growth formula: \(25,000 = 20,000 \cdot e^{0.05t}\).

Solve for t

To solve for t, we need to isolate the variable on one side of the equation. Start by dividing both sides of the equation by 20,000: \(\frac{25,000}{20,000} = e^{0.05t}\). Next, simplify the fraction on the left side of the equation: \(\frac{5}{4} = e^{0.05t}\). Now, we will take the natural logarithm (ln) on both sides of the equation to eliminate the exponential: \(ln\Big(\frac{5}{4}\Big) = ln\Big(e^{0.05t}\Big)\). Using a property of logarithms, we can rewrite the right side of the equation as: \(ln\Big(\frac{5}{4}\Big) = 0.05t \cdot ln(e)\). Since \(ln(e)\) is equal to 1, multiply both sides by \(\frac{1}{0.05}\) (or divide by 0.05) to obtain: \(t = \frac{ln\Big(\frac{5}{4}\Big)}{0.05}\).

Calculate the value of t

Plug in the values into the equation to solve for t: \(t = \frac{ln\Big(\frac{5}{4}\Big)}{0.05}\). Use a calculator to calculate the value of t: \(t \approx 4.32\).

Interpret the result

The result of t (4.32 years) suggests that it will take approximately 4.32 years for the acreage of wheat to increase from 20,000 acres to 25,000 acres, given a growth rate of 5% per year.