Question

a rectangle has an area of 72 in². the length and the width of the rectangle are changed by a scale factor of 3.5. what is the area of the new rectangle? enter your answer in the box. in²

Explanation:

Step1: Recall the formula for the area of a rectangle

The area of a rectangle is \(A = l\times w\), where \(l\) is the length and \(w\) is the width.

Step2: Analyze the effect of the scale factor on the area

If the length \(l\) is changed by a scale factor \(k = 3.5\) (new length \(l_{new}=k\times l\)) and the width \(w\) is changed by a scale factor \(k = 3.5\) (new width \(w_{new}=k\times w\)), then the new area \(A_{new}=l_{new}\times w_{new}=(k\times l)\times(k\times w)=k^{2}\times(l\times w)\).
Since \(A = l\times w = 72\space in^{2}\) and \(k = 3.5\), we substitute into the formula \(A_{new}=k^{2}\times A\).

Step3: Calculate the new area

\(A_{new}=(3.5)^{2}\times72\). First, calculate \((3.5)^{2}=3.5\times3.5 = 12.25\). Then, \(A_{new}=12.25\times72\).
\(12.25\times72=(12 + 0.25)\times72=12\times72+0.25\times72\).
\(12\times72 = 864\) and \(0.25\times72=\frac{1}{4}\times72 = 18\).
\(A_{new}=864 + 18=882\space in^{2}\).

Answer:

\(882\)