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

Answer:

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

Step-by-step explanation:

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

$y = -\frac{1}{8}x^2+4x$...........................................(1)

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

From (2),

$y = \frac{2}{3}x-\frac{14}{3}$.................................................(3)

Substituting (3) in (1),

$\frac{2}{3}x-\frac{14}{3} = -\frac{1}{8}x^2+4x$

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

$(24\times\frac{2}{3}x)-(24\times\frac{14}{3}) = (-24\times \frac{1}{8}x^2)+(24\times4x)$

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

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

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

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

$x=28$                       OR        $x = -\frac{4}{3}$

The value of y represents the height.

Substituting for x in (3)

$y = \frac{2}{3}\times28-\frac{14}{3} = \frac{56}{3} - \frac{14}{3} = 14$

$y = \frac{2}{3}\times(-\frac{4}{3})-\frac{14}{3} = -\frac{8}{9} - \frac{14}{3} = -\frac{50}{9}$

We discard the negative answer.

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