Question

1. a rectangular garden with length s and width 0.5s is surrounded on all four sides by a walkway of width 0.1s. if the width of the walkway is increased by 50 percent by expanding into the garden, what is the percent decrease in the area of the garden?

Explanation:

Step1: Calculate the original area of the garden

The area formula for a rectangle is \(A = length\times width\). Given the original length of the garden is \(s\) and the original width is \(0.5s\). So the original area \(A_{1}=s\times0.5s = 0.5s^{2}\).

Step2: Calculate the new width of the walk - way

The original width of the walk - way is \(0.1s\). If it is increased by \(50\%\), the new width of the walk - way \(w=0.1s\times(1 + 50\%)=0.1s\times1.5 = 0.15s\)

Step3: Calculate the new dimensions of the garden

The new length of the garden \(l_{2}=s-2\times0.15s=s - 0.3s = 0.7s\). The new width of the garden \(w_{2}=0.5s-2\times0.15s=0.5s - 0.3s=0.2s\)

Step4: Calculate the new area of the garden

Using the area formula \(A = length\times width\), the new area \(A_{2}=0.7s\times0.2s=0.14s^{2}\)

Step5: Calculate the percent decrease

The percent - decrease formula is \(\text{Percent Decrease}=\frac{A_{1}-A_{2}}{A_{1}}\times100\%\). Substitute \(A_{1}=0.5s^{2}\) and \(A_{2}=0.14s^{2}\) into the formula: \(\frac{0.5s^{2}-0.14s^{2}}{0.5s^{2}}\times100\%=\frac{0.36s^{2}}{0.5s^{2}}\times100\% = 72\%\)

Answer:

\(72\%\)