Question

find the dimensions of a rectangle whose perimeter is 30 meters and whose area is 56 square meters. the sides of the rectangle measure meters. (use a comma to separate answers as needed.)

Explanation:

Step1: Set up equations

Let the length be \( l \) and the width be \( w \).
Perimeter formula: \( 2(l + w)=30\), so \( l + w = 15\), then \( l=15 - w \).
Area formula: \( lw = 56\).

Step2: Substitute and solve

Substitute \( l = 15 - w \) into \( lw = 56\):
\((15 - w)w=56\)
\(15w-w^{2}=56\)
\(w^{2}-15w + 56=0\)
Factor the quadratic equation: \(w^{2}-15w + 56=(w - 7)(w - 8)=0\)
If \(w - 7=0\), then \(w = 7\); if \(w - 8=0\), then \(w = 8\).
When \(w = 7\), \(l=15 - 7 = 8\); when \(w = 8\), \(l=15 - 8 = 7\).

Answer:

\(7,8\)