Question

1. the length of a rectangle is nine inches less than twice the width. if the perimeter of the rectangle is 84 inches, find the dimensions of the rectangle. 11. find two consecutive integers whose sum is 93.

Explanation:

Question 10

Step1: Define variables

Let \( w \) be the width. Then length \( l = 2w - 9 \).
Perimeter formula \( P=2(l + w) \).

Step2: Substitute into perimeter formula

\( 84=2((2w - 9)+w) \)

Step3: Simplify equation

\( 84 = 2(3w - 9) \)
\( 84=6w - 18 \)

Step4: Solve for \( w \)

\( 6w=84 + 18 \)
\( 6w=102 \)
\( w = 17 \)

Step5: Find length

\( l=2(17)-9=34 - 9=25 \)

Step1: Define variables

Let the first integer be \( n \), then the next consecutive integer is \( n + 1 \).

Step2: Set up equation

\( n+(n + 1)=93 \)

Step3: Simplify equation

\( 2n+1 = 93 \)

Step4: Solve for \( n \)

\( 2n=93 - 1 \)
\( 2n=92 \)
\( n = 46 \)

Step5: Find the next integer

\( n + 1=46+1 = 47 \)

Answer:

Width is \( 17 \) inches and length is \( 25 \) inches.

Question 11