math

The equation \(y = -16t^2 - 18t + 405\) describes the height (in feet)

of a ball thrown downward at 18 feet per second from a height of 405 feet from the ground,

as a function of time t, in seconds.

In how many seconds will the ball hit the ground?

Express your answer as a decimal rounded to the nearest tenth.

\(\begin{array}{|rcll|} \hline y &=& -16t^2 - 18t + 405 \quad | \quad \text{the ball hit the ground at }~ y = 0 \\ 0 &=& -16t^2 - 18t + 405 \quad | \quad \times(-1) \\ 16t^2 + 18t - 405 &=& 0 \\\\ t &=& \dfrac{-18\pm \sqrt{18^2-4\times16\times(-405)} } {2\times 16} \\\\ t &=& \dfrac{-18\pm \sqrt{324+25920} } {32} \\\\ t &=& \dfrac{-18\pm \sqrt{26244} } {32} \\\\ t &=& \dfrac{-18\pm 162 } {32} \\\\ t &=& \dfrac{-18+162 } {32} \quad | \quad t > 0 ! \\\\ t &=& \dfrac{144} {32} \\\\ \mathbf{t} &=& \mathbf{4.5\ \text{seconds}} \\ \hline \end{array}\)