We have been given that part of the roof of a factory is devoted to mechanical support and part to green space. The area G that is designated as green space can be modeled by the polynomial [tex]2x^2-7x[/tex] and the area M that is devoted to mechanical support can be modeled by the polynomial [tex]x^2-9x+24.[/tex]
We are asked to find the area of the green space, when area of roof (R) is 36 square yards.
[tex]R=G+M[/tex]
[tex]R=2x^2-7x+x^2-9x+24[/tex]
[tex]36=2x^2-7x+x^2-9x+24[/tex]
Let us solve for x.
[tex]36=3x^2-16x+24[/tex]
[tex]3x^2-16x+24=36[/tex]
[tex]3x^2-16x+24-36=36-36[/tex]
[tex]3x^2-16x-12=0[/tex]
[tex]3x^2+2x-18x-12=0[/tex]
[tex]x(3x+2)-6(3x+2)=0[/tex]
[tex](3x+2)(x-6)=0[/tex]
[tex](3x+2)=0,(x-6)=0[/tex]
[tex]x=-\frac{2}{3},x=6[/tex]
Since length cannot be negative, therefore, the value of x would be 6.
The area of green space would be:
[tex]2x^2-7x\Rightarrow 2(6)^2-7(6)\\\\2x^2-7x=2(36)-42\\\\2x^2-7x=72-42\\\\2x^2-7x=30[/tex]
Therefore, the area of green space would be 30 square yards.
