Q13. - Algebra 1

Question

Juan hits a baseball. The equation $h=-16t^2+120t$ models the height $h$, in feet , of the ball after $t$ seconds. How long is the ball in the air?  

Step-by-Step Solution

Answer

For the duration 5.625 seconds the ball is in the air.

Step 1. Define an equation.

An equation is a mathematical expression that contains an equal symbol $"="$.

Step 2. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$ where, $a\ne 0$ is called the standard form of the quadratic function.

Step 3. Calculate the time for which the baseball is in the air.

The equation $h=-16t^2+120t$ represents the height $h$, in feet $t$, of the ball after seconds. To calculate the time duration for which the ball is in the air, $h$ should be equal to 0.

So, substitute $h=0$ in $h=-16t^2+120t$.

$$\begin{gathered} 0=-16t^2+120t \\ -8t\left(2t-15\right)=0 \\ t\left(2t-15\right)=\frac{0}{-8} \\ t\left(2t-15\right)=0 \end{gathered}$$

$t=0$ or $2t-15=0$

$t=0$ or $2t=15$

$t=0$ or $t=\frac{15}{2}$

$t=0$ or $t=7.5$

$t=0,7.5$

In this situation, these two points represent that the at the start $t=0$ seconds the ball is at the ground level and after $t=5.625$ seconds the ball will hit the ground level after traveling in the air.

Hence, for the duration of 5.625 seconds, the ball is in the air.