Question

1. the formula for the area a of a rectangle is $a = ell \times w$. the formula for the perimeter p of a rectangle is $p = 2ell + 2w$. the area of the front of a rectangular window flower box is shown.

$a = 10\frac{1}{2}$ square inches
$w = 2\frac{1}{4}$ inches
a. what is the length of the flower box?

b. what is the perimeter of the front of the flower box?

Explanation:

Part A: Length of the flower box

Step1: Recall the area formula for a rectangle

The area formula for a rectangle is \( A = \ell \times w \). We need to solve for \( \ell \), so we can rearrange the formula to \( \ell=\frac{A}{w} \).

Step2: Convert mixed numbers to improper fractions

Given \( A = 10\frac{1}{2}=\frac{2\times10 + 1}{2}=\frac{21}{2} \) square inches and \( w = 2\frac{1}{4}=\frac{4\times2+1}{4}=\frac{9}{4} \) inches.

Step3: Substitute values into the formula for length

Substitute \( A=\frac{21}{2} \) and \( w = \frac{9}{4} \) into \( \ell=\frac{A}{w} \), we get \( \ell=\frac{\frac{21}{2}}{\frac{9}{4}} \). When dividing fractions, we multiply by the reciprocal, so \( \ell=\frac{21}{2}\times\frac{4}{9} \).

Step4: Simplify the fraction

Simplify \( \frac{21}{2}\times\frac{4}{9}=\frac{21\times4}{2\times9}=\frac{84}{18}=\frac{14}{3}=4\frac{2}{3} \) inches.

Step1: Recall the perimeter formula for a rectangle

The perimeter formula for a rectangle is \( P = 2\ell+2w \). We already know \( \ell = 4\frac{2}{3}=\frac{14}{3} \) inches and \( w = 2\frac{1}{4}=\frac{9}{4} \) inches.

Step2: Calculate \( 2\ell \) and \( 2w \)

First, calculate \( 2\ell \): \( 2\times\frac{14}{3}=\frac{28}{3} \). Then, calculate \( 2w \): \( 2\times\frac{9}{4}=\frac{9}{2} \).

Step3: Add the two results

Now, find \( P=\frac{28}{3}+\frac{9}{2} \). To add these fractions, find a common denominator, which is 6. So \( \frac{28}{3}=\frac{56}{6} \) and \( \frac{9}{2}=\frac{27}{6} \). Then \( P=\frac{56}{6}+\frac{27}{6}=\frac{83}{6}=13\frac{5}{6} \) inches.

Answer:

The length of the flower box is \( 4\frac{2}{3} \) inches.

Part B: Perimeter of the front of the flower box