Example8.4.5

A physics class launches a tennis ball from a rooftop that is $$90.2$$ feet above the ground. They fire it directly upward at a speed of $$14.4$$ feet per second and measure the time it takes for the ball to hit the ground below. We can model the height of the tennis ball, $$h\text{,}$$ in feet, with the quadratic equation $$h=-16x^2+14.4x+90.2\text{,}$$ where $$x$$ represents the time in seconds after the launch. According to the model, when should the ball hit the ground? Round the time to one decimal place.

The ground has a height of $$0$$ feet. Substituting $$0$$ for $$h$$ in the equation, we have this quadratic equation:

\begin{equation*} 0=-16x^2+14.4x+90.2 \end{equation*}

We cannot solve this equation with factoring or the square root property, so we will use the quadratic formula. First we will identify that $$\highlight{a=-16}\text{,}$$ $$\highlight{b=14.4}$$ and $$\highlight{c=90.2}\text{,}$$ and substitute them into the formula:

\begin{align*} x\amp=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\ x\amp=\frac{-(\substitute{14.4})\pm\sqrt{(\substitute{14.4})^2-4(\substitute{-16})(\substitute{90.2})}}{2(\substitute{-16})}\\ x\amp=\frac{-14.4\pm\sqrt{207.36-(-5772.8)}}{-32}\\ x\amp=\frac{-14.4\pm\sqrt{207.36+5772.8}}{-32}\\ x\amp=\frac{-14.4\pm\sqrt{5980.16}}{-32}\\ \end{align*}

These are the exact solutions but because we have a context we want to approximate the solutions with decimals.

\begin{align*} x\amp\approx-2.0\text{ or }x\approx2.9 \end{align*}

We don't use the negative solution because a negative time does not make sense in this context. The ball will hit the ground approximately $$2.9$$ seconds after it is launched.

in-context