Question

Wheat growing is an example of

Short answer

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.