Question

A baseball is thrown up into the air. The ball's height in feet is modeled by the equation h(t) = -16t2 + 32t + 10. (h = height, t = time in seconds)
Find the maximum height of the ball.
a) 10 feet
b) 26 feet
c) 23 feet
d) 2.3 feet

Answer 1

Not sure if this is an optimization problem in your calculus class, so I'll answer from both an algebraic perspective and a calculus perspective.

Perspective 1: Algebra

your function gives the equation of a parabola of the form \[ax^2+bx+c\]
where a=-16, b=32, and c=10.

The crucial thing to recognize is that all parabolas are symmetric about the vertex. This means that if you find the x-intercepts, all you have to do is locate the point that is equidistant between them. this vertex will give you the maximum height. There is a simple formula for this (which is derived from the quadratic equation).

\[x_{vertex}=-b/(2a)=-32/(2*-16)=1\]
plugging this x value back into the original function gives the maximum height of \[-16+32+10=42-16=26\]
Thus, the maximum height of the ball is 26 ft.

Perspective 2: Calculus

This is the exact same thing as perspective 1 except that you use differentiation to find the vertex formula. The derivation is below.

set the equation of a parabola equal to zero
\[ax^2+bx+c=0\]
take the derivative
\[2*ax+b=0\]
solve for x
\[x=-b/(2*a)\]

And that's that :)

Answer 2

thank you