Question

determine the expression for the width of a rectangle
the area, a, of a rectangle is 120z² + 78z - 90, and the length, l, of the rectangle is 12z + 15. which of the following gives the width, w, of the rectangle?
10z - 19
10z - 6
8z - 6
9z + 4

Explanation:

Step1: Recall area formula

The area of a rectangle is $A = l\times w$. So, $w=\frac{A}{l}$.

Step2: Substitute given expressions

We have $A = 120z^{2}+78z - 90$ and $l=12z + 15$. Then $w=\frac{120z^{2}+78z - 90}{12z + 15}$.

Step3: Factor numerator and denominator

Factor out common factors: $120z^{2}+78z - 90 = 6(20z^{2}+13z - 15)$ and $12z + 15=3(4z + 5)$. So $w=\frac{6(20z^{2}+13z - 15)}{3(4z + 5)}=\frac{2(20z^{2}+13z - 15)}{4z + 5}$.
Factor $20z^{2}+13z - 15=(4z + 5)(5z-3)$. Then $w=\frac{2(4z + 5)(5z - 3)}{4z + 5}$.

Step4: Simplify the expression

Cancel out the common factor $(4z + 5)$ in the numerator and denominator. We get $w = 10z-6$.

Answer:

$10z - 6$