Question

the function $h(t) = -16t^2 + 100t + 5$ models the height of the ball from the ground at any given second, $t$, and is shown in the graph below.

graph of a parabola with x-axis labeled time (seconds) from -1 to 7 and y-axis labeled height of the ball (feet) from 0 to 160. the parabola starts near (0,0), rises to a peak around (3,160), then falls, crossing the x-axis near (6.3, 0).

estimate when $h(t) = 0$, and explain what it means in the given context.

Explanation:

Step1: Analyze the graph

The graph of \( h(t) = -16t^2 + 100t + 5 \) is a parabola opening downward. We need to find the time \( t \) when \( h(t) = 0 \), which is the x-intercept (other than the negative one, since time can't be negative in this context) of the parabola.
From the graph, we can see that the parabola intersects the t - axis (where \( h(t)=0 \)) at \( t\approx - 0.05 \) (which we can ignore as time can't be negative) and \( t\approx6.3 \) (by estimating from the graph, looking at the x - axis where the height is 0).

Step2: Interpret the context

In the context of the height of the ball, \( h(t) = 0 \) means that the height of the ball from the ground is 0 feet. So, when \( t\approx6.3 \) seconds, the ball hits the ground (since its height above the ground is 0).

Answer:

\( t\approx6.3 \) seconds (this means the ball hits the ground at approximately 6.3 seconds, as its height from the ground is 0 at this time).