- The area of the corn field is modeled by the equation \(A(t) = 80,000\left(\frac{3}{4}\right)^t\).
- Setting this equal to the target area, the equation is \(80,000\left(\frac{3}{4}\right)^t = 33,750\).
- Solving \(\left(\frac{3}{4}\right)^t = \frac{33,750}{80,000} = \frac{27}{64}\) reveals that \(\left(\frac{3}{4}\right)^t = \left(\frac{3}{4}\right)^3\).
- The final result is \(t = 3\) months.
- The correct equation is \(80,000\left(\frac{3}{4}\right)^t=33,750\) and the time is 3 months.
Explanation
- Formulating the Decay Equation
We are given an initial corn field area \(A_0 = 80,000\) square feet. Each month, the area decreases by one-fourth, meaning the area becomes \(\left(1 - \frac{1}{4}\right) = \frac{3}{4}\) of its previous size at the end of every month. The area after \(t\) months can be modeled by the exponential decay formula:
\[A(t) = A_0 \cdot \left(\frac{3}{4}\right)^t = 80,000 \cdot \left(\frac{3}{4}\right)^t\]
Our goal is to find the number of months \(t\) such that the area \(A(t) = 33,750\) square feet. Setting these equal, we get the equation:
\[80,000 \left(\frac{3}{4}\right)^t = 33,750\]
This corresponds to one of the options provided in the equation table.
- Solving for Time \(t\)
To find the value of \(t\), we divide both sides by \(80,000\):
\[\left(\frac{3}{4}\right)^t = \frac{33,750}{80,000}\]
Simplifying the fraction \(\frac{33,750}{80,000}\) by dividing the numerator and denominator by their greatest common divisor (which is \(1,250\)):
\[\frac{33,750 \div 1,250}{80,000 \div 1,250} = \frac{27}{64}\]
Now the equation is:
\[\left(\frac{3}{4}\right)^t = \frac{27}{64}\]
Recognizing that \(27 = 3^3\) and \(64 = 4^3\), we can write the right side as a power of \(\frac{3}{4}\):
\[\left(\frac{3}{4}\right)^t = \left(\frac{3}{4}\right)^3\]
Comparing the exponents, we find that \(t = 3\). Thus, it will take 3 months for the field to reach the target area.
Examples
Exponential decay is a crucial concept in many real-life applications. For example, radioactive isotopes have a "half-life," which is the time it takes for half of the substance to decay. By using the same mathematical principles applied here, scientists can calculate how much of a radioactive material will remain after a specific amount of time or determine the age of ancient artifacts through carbon dating. Similar models are also used in finance to track the depreciation of assets over time, helping businesses manage their budgets and plan for future equipment replacements. Understanding these formulas allows for accurate prediction and resource management in various scientific and professional fields.