Question

persevere if the area of a rectangle is 144\sqrt{5} square inches, what are possible dimensions of the rectangle? explain your reasoning. because 12\sqrt{12}\times6\sqrt{15}=144\sqrt{5}. 12\sqrt{12} in. is a possible length and select choice\sqrt{15} in. is a possible width. need help with this question?

Explanation:

Step1: Recall area formula for rectangle

The area of a rectangle is \(A = l\times w\), where \(l\) is the length and \(w\) is the width, and \(A = 144\sqrt{5}\).

Step2: Analyze the given - incorrect - multiplication

The expression \(12\sqrt{12}\times6\sqrt{15}\) is calculated as follows: First, multiply the non - square - root parts: \(12\times6 = 72\). Then multiply the square - root parts: \(\sqrt{12}\times\sqrt{15}=\sqrt{12\times15}=\sqrt{180}=\sqrt{36\times5}=6\sqrt{5}\). So \(12\sqrt{12}\times6\sqrt{15}=72\times6\sqrt{5}=432\sqrt{5}
eq144\sqrt{5}\).

Step3: Find correct factors

We know that \(144\sqrt{5}\) can be written as a product of two numbers. Let's assume one side is \(12\sqrt{12}\), and we want to find the other side \(x\) such that \(12\sqrt{12}\times x=144\sqrt{5}\). Then \(x=\frac{144\sqrt{5}}{12\sqrt{12}}\). Simplify \(\frac{144}{12} = 12\), and \(\frac{\sqrt{5}}{\sqrt{12}}=\sqrt{\frac{5}{12}}\). Another way is to factorize \(144\sqrt{5}\) as \(12\sqrt{12}\times4\sqrt{15}\) because \(12\times4 = 48\) and \(\sqrt{12}\times\sqrt{15}=\sqrt{180}=6\sqrt{5}\), and \(48\times3\sqrt{5}=144\sqrt{5}\).

Answer:

The correct value for the width is \(4\sqrt{15}\) inches when the length is \(12\sqrt{12}\) inches. So the answer for the "Select Choice" is 4.