Question

a preschool has a rectangular field and a rectangular playground that are similar in shape. each dimension of the field is 3.2 times the corresponding dimension of the playground. which statement is true? the area of the field is 6.4 times the area of the playground. the area of the field is 10.24 times the area of the playground. the perimeter of the field is 6.4 times the perimeter of the playground. the perimeter of the field is 10.24 times the perimeter of the playground.

Explanation:

Step1: Recall the formulas for perimeter and area of a rectangle

Let the length and width of the playground be \(l\) and \(w\). Then the perimeter of the playground \(P_{1}=2(l + w)\), and the area of the playground \(A_{1}=lw\). The length and width of the field are \(3.2l\) and \(3.2w\).

Step2: Calculate the perimeter of the field

The perimeter of the field \(P_{2}=2(3.2l+3.2w)=3.2\times2(l + w)\). So, \(\frac{P_{2}}{P_{1}}=\frac{3.2\times2(l + w)}{2(l + w)} = 3.2\).

Step3: Calculate the area of the field

The area of the field \(A_{2}=(3.2l)\times(3.2w)=3.2\times3.2lw\). Since \(3.2\times3.2 = 10.24\), \(\frac{A_{2}}{A_{1}}=\frac{3.2\times3.2lw}{lw}=10.24\).

Answer:

The area of the field is 10.24 times the area of the playground.