Option A
[tex]4 x^{2}+32 x-80=0[/tex] is the required equation to calculate width of rectangular frame that has a total area of 140 square inches.
Solution:
Given that,
Length of a rectangular frame is given as 2x + 10
Width of the rectangular frame is given as 2x + 6
Total area = 140 square inches
The area of rectangular frame is given as:
[tex]\text {area of rectangle}=\text {length } \times \text {width}[/tex]
Plugging in values, we get
[tex]\begin{array}{l}{(2 x+10)(2 x+6)=140} \\\\ {4 x^{2}+12 x+20 x=80} \\\\ {4 x^{2}+32 x-80=0}\end{array}[/tex]
This is the required equation to calculate width of rectangular frame
Solve the above quadratic equation to get the value of "x"
[tex]4 x^{2}+32 x-80=0[/tex]
Use the quadratic equation formula:
[tex]x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex]
Here a = 4 ; b = 32 ; c = -80
[tex]x=\frac{-32 \pm \sqrt{32^{2}-4(4)(-80)}}{2(4)}[/tex]
[tex]x=\frac{-32 \pm \sqrt{2304}}{8}[/tex]
x = 2 or x = -10
Now measurement cannot be negative, so taking the positve value of "x", we can calculate the width
So put "x" = 2
Width of the rectangular frame = 2x + 6 = 2(2) + 6 = 10
Thus the width of frame is 10 inches
