A rectangular city park is $\frac{6}{8}\text{ mi}$ long. The park has an area of $\frac{1}{4}\text{ mi}^2$. What is the width of the park? Show your work.
The area \(A\) of a rectangle is calculated by multiplying its length \(L\) and its width \(W\), expressed as:
\[A = L \times W\]
In this problem, we are given:
Our objective is to determine the width \(W\).
First, simplify the given length of the park. The fraction \(\frac{6}{8}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is \(2\):
\[L = \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \text{ mi}\]
Now that we have the area and the simplified length, we use the area formula to set up the equation for the width:
\[\frac{1}{4} = \frac{3}{4} \times W\]
To solve for the width \(W\), isolate the variable by multiplying both sides of the equation by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\):
\[W = \frac{1}{4} \times \frac{4}{3}\]
Performing the multiplication:
\[W = \frac{1 \times 4}{4 \times 3} = \frac{4}{12}\]
Simplifying the resulting fraction by dividing the numerator and denominator by \(4\):
\[W = \frac{1}{3} \text{ mi}\]
The width of the park is \(\frac{1}{3} \text{ mi}\).
Understanding the relationship between area, length, and width is essential for land management and urban planning. For example, if a city planner needs to design a recreational area within a specific land parcel, they must calculate the necessary width for a park based on the available area and the required length to fit it into the city landscape. Using the formula \(W = A / L\), the planner ensures that the final design meets all spatial requirements effectively and precisely.