Suppose the height, h, in feet, of a trampolinist above the ground during one bounce is modelled by the quadratic function h(t) = -16t^2 + 42t + 3.75 . For what period of time is the trampolinist at least 22 ft above the ground? Round your answers to the nearest hundredth. (2 marks)
http://www.wolframalpha.com/input/?i=-16t^2+%2B+42t+%2B+3.75+%3E%3D+22
To determine the period of time during which the trampolinist is at least 22 ft above the ground, we need to solve the inequality h(t) ≥ 22.
Given the quadratic function h(t) = -16t^2 + 42t + 3.75, we can rewrite the inequality as follows:
-16t^2 + 42t + 3.75 ≥ 22
To solve this quadratic inequality, we first set it equal to zero by subtracting 22 from both sides:
-16t^2 + 42t + 3.75 - 22 ≥ 0
Simplifying this equation, we get:
-16t^2 + 42t - 18.25 ≥ 0
To solve this quadratic inequality, we can use algebraic techniques or graphing methods. In this case, let's solve it algebraically:
First, find the roots of the quadratic equation -16t^2 + 42t - 18.25 = 0:
Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a)
where a = -16, b = 42, and c = -18.25
t = (-42 ± √(42^2 - 4(-16)(-18.25))) / (2(-16))
t = (-42 ± √(1764 + 1172)) / (-32)
t = (-42 ± √(2936)) / (-32)
Now, calculate the two roots:
t = (-42 + √(2936)) / (-32) ≈ -0.14
t = (-42 - √(2936)) / (-32) ≈ 2.89
Since the trampolinist's height is always above the ground during one bounce, we can ignore the negative root.
So, the trampolinist is at least 22 ft above the ground in the time interval (0.14, 2.89). Rounded to the nearest hundredth, the trampolinist is at least 22 ft above the ground for approximately 0.14 to 2.89 seconds.