Question

The height of a child’s bounce in feet above a trampoline is given by the function h(t) = -16t2 + 48t - 20. Find the maximum height of the child’s bounce.

Answer 1

$\large\textsf{Quadratic equation:}~~\fbox{ \mathsf{ax^2+bx+c} }\\\\\\\mathsf{When~a\ \textless \ 0~the~parable~have~concavity~\underline{downward}~and~have~one~max}\\\mathsf{point.}\\\\\\\mathsf{The~max~point~is~a~vertex~of~the~parable~and~its~coordinates~are}\\\mathsf{obteined~by:}\\\\\\\mathsf{x~of~vertex~X_v=\dfrac{-b}{2a}}\\\\\\\mathsf{y~of~vertex~Y_v=~\dfrac{-\Delta}{4a}}\\\\\\\mathsf{For~this~question~we~must~find~the~Y_v.}$


$\mathsf{-16t^2+48t-20}\\\\\\\mathsf{coefficient~a=-16;}\\\mathsf{coefficient~b=48;}\\\mathsf{independent~term=-20.}$


$\mathsf{Y_v=\dfrac{-\Delta}{4a}}\\\\\\\mathsf{\Delta=b^2-4ac}\\\\\\\mathsf{\Delta=48^2-4\cdot(-16)\cdot(-20)}\\\\\\\mathsf{\Delta=2304-1280}\\\\\\\mathsf{\Delta=1024}\\\\\\\mathsf{Y_v=\dfrac{-1024}{4\cdot(-16)}}\\\\\\\mathsf{Y_v=\dfrac{-1024}{-64}}\\\\\\\fbox{ \mathsf{Yv=16} }~~~~\checkmark$

$\textsf{So~the~maximum~height~of~the~child's~bounce~is~16.}$