Question

the function $h(t) = -3t^2 + 84t$ gives the height (in feet) of a golf ball after $t$ seconds. use $h(t)$ to answer the questions below. round answers to two decimal places as needed. verify your answer by graphing.
use factoring to determine when $h(t) > 0$. write your answer in interval notation and explain your answer by completing the sentence below.
factored form: $square > 0$
$h(t) > 0$ over the interval $square$
the height of the ball is greater than 0 from $square$ to $square$ seconds

Explanation:

Step1: Factor the function

Given \( H(t) = -3t^2 + 84t \), factor out the greatest common factor, which is \(-3t\). So we get \( H(t)=-3t(t - 28) \). To find when \( H(t)>0 \), we set up the inequality \(-3t(t - 28)>0\).

Step2: Analyze the inequality

First, note that the coefficient of \( t^2 \) is negative (\(-3\)), so the parabola opens downward. The roots of the equation \( H(t) = 0 \) are found by setting each factor equal to zero: \( -3t=0\) gives \( t = 0 \), and \( t - 28=0\) gives \( t=28 \). For a downward - opening parabola, the function \( y = H(t) \) is positive between its two roots. So we solve the inequality \(-3t(t - 28)>0\). Since \(-3<0\), the inequality is equivalent to \( t(t - 28)<0\) (when we divide both sides of an inequality by a negative number, the direction of the inequality sign changes). The solution to \( t(t - 28)<0\) is \( 00 \) and \( t - 28<0\) (i.e., \( t<28 \)), then \( 00\) (i.e., \( t>28 \)), there is no solution for \( t \) that satisfies both conditions.

Answer:

Factored form: \(-3t(t - 28)>0\)
\(H(t)>0\) over the interval \((0, 28)\)
The height of the ball is greater than 0 from \(0\) to \(28\) seconds.