Question

At a basketball game, an air cannon launches T-shirts into the crowd. The function y=−1/8x^2 + 4x represents the path of a T-shirt. The function 3y =2x − 14 represents the height of the bleachers. In both functions, y represents vertical height (in feet) and x represents horizontal distance (in feet). At what height does the T-shirt land in the bleachers?

Answer 1

The T-shirt lands in the bleachers at a height of 14 feet.

Explanation:

The point where they meet is the solution of the simultaneous system of equations:

`y = -(1)/(8)x^2+4x`...........................................(1)

`3y = 2x - 14`..............................................(2)

From (2),

`y = (2)/(3)x-(14)/(3)`.................................................(3)

Substituting (3) in (1),

`(2)/(3)x-(14)/(3) = -(1)/(8)x^2+4x`

Multiply through by 24 (the LCM of 3 and 8, the denominators of the fractions)

`(24*(2)/(3)x)-(24*(14)/(3)) = (-24* (1)/(8)x^2)+(24*4x)`

`16x - 112 = -3x^2+96x`

`3x^2-80x -112 = 0`

`(x-28)(3x+4) = 0`

`x-28 = 0` OR
`3x+4=0`

`x=28` OR
`x = -(4)/(3)`

The value of y represents the height.

Substituting for x in (3)

`y = (2)/(3)*28-(14)/(3) = (56)/(3) - (14)/(3) = 14`

`y = (2)/(3)*(-(4)/(3))-(14)/(3) = -(8)/(9) - (14)/(3) = -(50)/(9)`

We discard the negative answer.

Hence, the T-shirt lands in the bleachers at a height of 14 feet.