Suppose the height, h, in feet, of a trampolinist above the ground during one bounce is modeled 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)

To find the period of time during which the trampolinist is at least 22 ft above the ground, we need to determine the values of 't' that satisfy the condition h(t) ≥ 22.

First, let's rewrite the quadratic equation:

h(t) = -16t^2 + 42t + 3.75

Now, set h(t) equal to 22 and solve for 't':

-16t^2 + 42t + 3.75 = 22

Rearranging the equation:

-16t^2 + 42t + 3.75 - 22 = 0

Combine like terms:

-16t^2 + 42t - 18.25 = 0

Now, we can use the quadratic formula to find the values of 't'. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -16, b = 42, and c = -18.25.

Plugging these values into the quadratic formula:

t = (-42 ± √(42^2 - 4(-16)(-18.25))) / (2(-16))

Simplifying:

t = (-42 ± √(1764 - 1168)) / (-32)

t = (-42 ± √596) / (-32)

Now, calculate the two possible values for 't':

t₁ = (-42 + √596) / (-32)
t₂ = (-42 - √596) / (-32)

Using a calculator, we find:

t₁ ≈ 1.92
t₂ ≈ -0.32

The trampolinist cannot have a negative time value, so we discard t₂.

Therefore, the trampolinist is at least 22 ft above the ground for a period of approximately 1.92 seconds.