Explanation:
the flight path of a rocket or a thrown object is a parabola as described by the function.
h(t) = -16t² + 40t + 7
the negative sign of the prime coefficient (-16) tells us that the parabola opens downwards, and that the vertex as the extreme point of the parabola is a maximum.
we get the extreme point of a curve by finding the zero solution of the first derivative of the function.
and the extreme value itself by using that value in the original equation and calculate the result.
h'(t) = -32t + 40
0 = -32t + 40
32t = 40
t = 40/32 = 5/4 = 1.25 seconds
h(5/4) = -16×(5/4)² + 40×5/4 + 7 =
= -16×25/16 + 10×5 + 7 =
= -25 + 50 + 7 = 32 ft
the rocket will reach its maximum height of 32 ft after 1.25 seconds.