Question

The function $h(x)=-16x^2 + 48x + 6$ models the height $h$ (in feet) of a ball ejected from a ball launcher after $x$ seconds. When does the ball reach a height of 30 feet? If necessary, round to the nearest thousandth.

A) $x = 0.750$
B) $x = 1.500$
C) $x = 2.250$
D) $x = 3.000$

Answer 1

The ball reaches a height of 30 feet at approximately 0.750 seconds after launch, which corresponds to Option A.

Step-by-step explanation:

We're giving the quadratic function h(x) = -16x^2 + 48x + 6, which models the height of a ball after it's launched. To find out when the ball reaches a height of 30 feet, we set the equation equal to 30 and solve for x:

30 = -16x^2 + 48x + 6

Moving everything to one side gives us the quadratic equation:

0 = -16x^2 + 48x - 24

Dividing through by -16 to simplify, we get:

0 = x^2 - 3x + 1.5

This is a quadratic equation and we can use the quadratic formula, x = (-b ± √(b^2-4ac))/(2a), to find the roots. After applying the formula, we find:

x = (3 ± √(9 - 6))/(2)

Thus, there are two times where the ball reaches 30 feet, but we are looking for the earliest time when the height is ascending. Only one of these solutions will be a positive time near one of the options given and that is x = 0.750 seconds, which corresponds to Option A.