Question

write expressions for the perimeter and area for the figure. then simplify these expressions. assume that all measures are given in inches. the perimeter is ( 20sqrt{5} ) inches. (simplify your answer. type an exact answer, using radicals as needed.) the area is ( square ) square inches. (simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Simplify \(\sqrt{125}\)

We know that \(125 = 25\times5\), so \(\sqrt{125}=\sqrt{25\times5}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 25\), \(b = 5\)), we get \(\sqrt{125}=\sqrt{25}\cdot\sqrt{5}=5\sqrt{5}\).

Step2: Find the perimeter formula and calculate

The perimeter \(P\) of a rectangle is \(P = 2(l + w)\), where \(l=\sqrt{125}+4\) and \(w=\sqrt{125}-4\). Substitute \(l\) and \(w\) into the formula:

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Since \(\sqrt{125}=5\sqrt{5}\), then \(P = 2\times2\times5\sqrt{5}=20\sqrt{5}\) (this is already given, just for verification).

Step3: Find the area formula and calculate

The area \(A\) of a rectangle is \(A=l\times w\), where \(l=\sqrt{125}+4\) and \(w=\sqrt{125}-4\). Using the formula \((a + b)(a - b)=a^{2}-b^{2}\) (\(a=\sqrt{125}\), \(b = 4\)):

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Answer:

The area is \(109\) square inches.