Question

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 2x2 - 7x and the area M that is devoted to mechanical support can be modeled by the polynomial x2 - 9x + 24. Given that the area R of the roof is 36 square yards, write and solve a quadratic equation to find the total area of the green space. **Use the positive value for your solution.

(Hint: R = G + M)

Answer 1

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
`2x^2-7x` and the area M that is devoted to mechanical support can be modeled by the polynomial
`x^2-9x+24.`

We are asked to find the area of the green space, when area of roof (R) is 36 square yards.

`R=G+M`

`R=2x^2-7x+x^2-9x+24`

`36=2x^2-7x+x^2-9x+24`

Let us solve for x.

`36=3x^2-16x+24`

`3x^2-16x+24=36`

`3x^2-16x+24-36=36-36`

`3x^2-16x-12=0`

`3x^2+2x-18x-12=0`

`x(3x+2)-6(3x+2)=0`

`(3x+2)(x-6)=0`

`(3x+2)=0,(x-6)=0`

`x=-(2)/(3),x=6`

Since length cannot be negative, therefore, the value of x would be 6.

The area of green space would be:

`2x^2-7x\Rightarrow 2(6)^2-7(6)\\\\2x^2-7x=2(36)-42\\\\2x^2-7x=72-42\\\\2x^2-7x=30`

Therefore, the area of green space would be 30 square yards.