Problema Solution
Before a basketball game, a referee practices tossing the ball vertically into the air for the tip-off. The ball's height(h which is measured in feet) after a certain number of seconds (t) is given by h=-16t^2+24t+6. How long does it take for the ball to hit the floor?
Use the quadratic formula to see at what time(s) would the ball be at a height of nine feet?
What is the maximum height reached by the ball? Hint- think vertex
Answer provided by our tutors
the height of the ball is given by
h(t) = - 16t^2 + 24t + 6
if we draw the graph of this function we will see that it is parabolic with vertex (0.75, 15)
click here to see the graph
How long does it take for the ball to hit the floor?
this means we have to find 't' such that h(t) = 0 that is
- 16t^2 + 24t + 6 = 0 solve this quadratic equation and consider the positive roots
click here to see the step by step solution
t = (3 + (15^(1/2))/4
t = 1.72 s
the ball needs 1.72 seconds to hit the floor
What time would the ball be at a height of nine feet?
we need to find t such that h(t) = 9 ft that is solve the quadratic equation
- 16t^2 + 24t + 6 = 9
click here to see the step by step solution
we find 2 solutions
t1 = (3 + 6^(1/2))/4
t2 = (3 - 6^(1/2))/4
or
t1 = 0.14 s
t2 = 1.36 s
after 0.14 seconds and 1.13 seconds the ball will be at 9 feet height.
What is the maximum height reached by the ball?
the vertex of the function h(t) = - 16t^2 + 24t + 6 gives the maximum heights
look at the graph and you can easily spot the vertex (0.75, 15)
or using the formula for the vertex
t = - b / 2a = -24/-32 = 0.75
h (0.75) = 15 ft
the maximum height is 15 ft.