Question

The local gym is putting in a beach volleyball court! They sawthat the standard size for a court is 30 feet by 60 feet, so that iswhat they made. However, they realized they should have gonebigger in each direction to give some boundary sand. They haveenough sand to cover a total area of 4536 square feet.a) Set up an equation to solve for how far they can expandthe court in each direction (include a picture):b) Solve your above equation and include units in youranswer; how far can they expand the court in eachdirection?

Answer 1

The diagram illustrating the scenario is shown below

a) The inner box represents the initial size of the court that they made. x represents the extra distance that they could have added on either sides so that the area would be 4536 square feet. Recall, the area of a rectangle is length x width

Thus,

the new length = 60 + x + x = 60 + 2x

the new width = 30 + x + x = 30 + 2x

area = 4536

thus, the equation is

(60 + 2x)(30 + 2x) = 4536

b) We would expand the brackets. it becomes

60 * 30 + 60 * 2x + 2x * 30 + 2x * 2x = 4536

1800 + 120x + 60x + 4x^2 = 4536

4x^2 + 120x + 60x + 1800 = 4536

4x^2 + 180x + 1800 = 4536

Dividing both sides of the equation by 4, we have

x^2 + 45x + 450 = 1134

x^2 + 45x + 450 - 1134 = 0

x^2 + 45x - 684 = 0

We would solve the quadratic equation by factorising. We would find two terms such that their sum or difference is 45x and their product is - 684x^2. The terms are 57x and - 12x. thus, we have

x^2 + 57x - 12x - 684 = 0

By factorising, we have

x(x + 57) - 12(x + 57) = 0

(x - 12)(x + 57) = 0

x - 12 = 0 or x + 57 = 0

x = 12 or x = - 57

The value of x cannot be negative. Thus, x = 12

Therefore

The length and width of the court can be expanded by

12 * 2 = 24 feet

new length = 60 + 24 = 84 ft

new width = 30 + 24 = 54 ft

Checking, 84 x 54 = 4536 square feet