Question

If a projectile is fired straight upward from the ground with an initial speed of 160 feet per​ second, then its height h in feet after t seconds is given by the function ​h(t)=−16t2+160t. Find the maximum height of the projectile.

Answer 1

Explanation:

the function is

h(t) = -16t² + 160t

this is a parabola of the form

ax² + bx + c

with

a = -16

b = 160

c = 0

since a < 0, we know the parabola is opening downwards (as expected for the flight curve of an object).

and that means that the vertex of the parabola is the highest point in the flight curve of the projectile.

and so, of the vertex (h, k), the k-coordinate (the y-value of the point) is the maximum height.

we know that

h = -b/(2a) = -160/(2×-16) = -160/-32 = 10/2 = 5

so, we know that the projectile reaches its highest point after 5 seconds of flight.

and so, that highest point is

k = h(5) = -16×5² + 160×5 = -16×25 + 800 = -400 + 800 = 400 ft

the maximum height is 400 ft.