Eli is putting a cement slab in his backyard the original slab was going to have dimensions of 8 ft by 6 ft he decided to make the slab larger by adding x feet to each side the area of the new slab is now 120 sq ft

a.Write quadratic equation that represents the area of the new slab.
b. Find the new dimensions of the slab.

1 answer

To solve the problem, we can follow the steps outlined below:

Original Dimensions: The original dimensions of the slab are 8 ft by 6 ft.
New Dimensions: If Eli adds \( x \) feet to each side, the new dimensions of the slab will be:
- Length: \( 8 + x \) ft
- Width: \( 6 + x \) ft
Area of New Slab: The area of the new slab is given as 120 sq ft. Therefore, the equation for area can be set up as follows: \[ (8 + x)(6 + x) = 120 \]
Expanding the Equation: \[ 48 + 8x + 6x + x^2 = 120 \] \[ x^2 + 14x + 48 = 120 \]
Setting the Quadratic Equation: Rearranging this equation gives: \[ x^2 + 14x + 48 - 120 = 0 \] \[ x^2 + 14x - 72 = 0 \]

Thus, the quadratic equation representing the area of the new slab is: \[ x^2 + 14x - 72 = 0 \]

Solving the Quadratic Equation: We can solve for \( x \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 14 \), and \( c = -72 \).
Calculating the Discriminant: \[ b^2 - 4ac = 14^2 - 4 \times 1 \times (-72) = 196 + 288 = 484 \]
Finding \( x \): \[ x = \frac{-14 \pm \sqrt{484}}{2 \times 1} = \frac{-14 \pm 22}{2} \]
- Calculating the Two Possible Solutions: \[ x = \frac{8}{2} = 4 \quad (\text{since } -18 \text{ is not a valid length}) \]
New Dimensions:
- Length: \( 8 + x = 8 + 4 = 12 \) ft
- Width: \( 6 + x = 6 + 4 = 10 \) ft

Therefore, the new dimensions of the slab are: