Applications of Quadratic Equations and Functions

Question

A soccer ball is kicked into the air so that its height, h, in feet, after t seconds is given by the

function:

h(t) = −16t²t+ 96

What is the maximum height the ball reaches?

How long is the ball in the air?

Answer 1

You may have mistyped your equation, because that equation doesn't look correct. Fortunately, I can still guide you on how to solve this:

Quadratic equations look like a parabola. If the coefficient of the x² term is positive, then the parabola opens up, so it has a minimum value (the lowest point in the graph). And if the coefficient is negative -- as in the case of this problem -- then the parabola opens down, so it has a maximum value (the highest point in the graph).

To find the maximum in this problem, find the axis of symmetry using the formula x = -b/(2a). Plug what you get back into h(t) to get the maximum height the ball reaches. (note this method also works for finding minimums too.)

How long the ball is in the air is equivalent to how long h(t) is positive. To do this, find the x-intercepts of h(t), and take their difference.

Answer 2

h(t)=-16t^2 +96 has maximum height = 96 feet

but that makes no sense

It's likely

it should have read:

h(t)=-16t^2 +96t

then take the derivative and set =0

h'(t) = v(t) = -32t +96= 0

t = 96/32 = 3 seconds to max height

max height = h(3) = -16(3)^2 +96(3) = -144+288=144 feet

it's in the air when

h(t)=-16t^2 + 96t = 0

-16t(t-6)=0

from t= 0 to t=6, for 6 seconds until it hits the ground

that's twice the time to reach maximum height

if you have a graphing calculator, plug in the equation and find the vertex and x intercepts. the positive x intercept is the splash down. the vertex gives the maximum height, as its a downward opening parabola

rewrite -16t^2 + 96t in vertex and x intercept forms

-16t^2 + 96t

= -16(t^2 -96/16 + (96/32)^2) + 16(96/32)^2

= -16(t - 3)^2 + 144

vertex is (3,144) 144 ft is max height

-16t^2 +96t

=-16t(t-3)

t-3=0

t=3

x intercepts are 0 and 3 seconds to reach max height

twice 3 = 6 seconds to reach splashdown