Question

1. the formula for the perimeter p of a rectangle with length ε and width w is

p = 2(ε + w). find the length of the rectangle shown here if the perimeter is 9\frac{1}{2} feet.
solve the equation for ε and justify each step.

Explanation:

Step1: Substitute the values into the formula

Given \( P = 9\frac{1}{2}=\frac{19}{2}\) feet and \( w=\frac{1}{2}\) foot. The formula \( P = 2(\ell+w)\). Substitute \(P\) and \(w\) into the formula: \(\frac{19}{2}=2(\ell+\frac{1}{2})\).

Step2: Divide both sides by 2

By the division - property of equality, \(\frac{\frac{19}{2}}{2}=\ell+\frac{1}{2}\). Since \(\frac{\frac{19}{2}}{2}=\frac{19}{2}\times\frac{1}{2}=\frac{19}{4}\), the equation becomes \(\frac{19}{4}=\ell+\frac{1}{2}\).

Step3: Subtract \(\frac{1}{2}\) from both sides

Using the subtraction - property of equality, \(\ell=\frac{19}{4}-\frac{1}{2}\). Rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\). Then \(\ell=\frac{19 - 2}{4}=\frac{17}{4}=4\frac{1}{4}\) feet.

Answer:

The length \(\ell\) is \(4\frac{1}{4}\) feet.