Question

A water balloon is launched upward at the rate of 86 ft/sec. Using the formula $h=-16t^2+86t$, find how long it will take the balloon to reach the maximum height and then find the maximum height. Round to the nearest tenth.

Short answer

The maximum height occurs when $t=2.7$seconds and the maximum height is $115.56$feet.

Step-by-step solution

Given information

The given quadratic equation is $h=-16t^2+86$.

Calculate how long it will take the balloon to reach maximum height.

Compare the given equation with standard form $f\left(x\right)=a{x}^2+bx+c$we get $a=-16,b=86andc=0$.

Since the leading coefficient is negative so parabola opens down and the quadratic function has a maximum.

$$\begin{gathered} t=-\frac{b}{2a} \\ =-\frac{86}{2(-16)} \\ =\frac{86}{32} \\ \approx 2.7 \end{gathered}$$

The maximum occurs when$t=2.7s$.

Calculate the maximum height.

The height will be maximum when $t=2.7$ so we will put $t=2.7$into the given equation.

$$\begin{gathered} h(2.7)=-16{(2.7)}^2+86(2.7) \\ =-116.64+232.2 \\ =115.56 \end{gathered}$$

The maximum height is $115.56$feet.