Question

A ball is thrown into the air from an upstairs window and its height, h, in feet t seconds after it is thrown is given by the equation: h = –( t^2 – 8t – 20)

a. What is the maximum height that the ball reaches? You can check your answer with a graphing utility, but first try to determine your answer without the graph by rewriting the equation in vertex form and using the information in the rewritten form.

b. After how many seconds will it reach the ground? You can check your answer with a graphing utility, but first try using the equation to determine your answer by rewriting the equation in factored form.

Answer 1

The height, h, in feet, of a ball thrown into the air from an upstairs window, t seconds after it is thrown, is given by the equation h= –( t²– 8t-20).

1. We have h = –( t²– 8t-20)=-(t²-2t*4 +4²) +20+ 4² = -(t-4)² +36. This is the vertex form of the equation of an upwards opening parabola with vertex at (4,36). Since the vertex is the highest point of such a parabola, hence the maximum height that the ball reaches, is 36 ft. A graph of the given equation is attached.
2. When the ball reaches the ground, h = 0 or, t²– 8t-20 = 0 or, (t-10)(t+2) = 0 or, t- 10=0 ( as t cannot be negative). Thus, the ball will reach the ground after 10 seconds. It may be observed from the graph that the parabola crosses the X-Axis ( or, rather, the t-Axis) at t = 10