Finding the Highest Point

Date: 4/13/96 at 21:47:9
From: Anonymous
Subject: Story Problems, very confused.


How would you solve this problem?: 

A ball is thrown into the air, and its height, h, at any time, t, 
in seconds is given by h = -16(t-1)^2 = 212. When is the ball at 
its highest point above ground? When is it on the ground? 

See why I'm so confused? 

Date: 4/17/96 at 15:41:34
From: Doctor Aaron
Subject: Re: Story Problems, very confused.

Hi,

I think that you made a typo.  If h = -16(t-1)~2 = 212, then 
h = 212 and we have a specific value of t, so it's like the ball 
is frozen in time. I'll assume that the = is a + because that 
makes a lot more sense.

I don't know how much background you have, but I'll start with 
when the ball is on the ground.  When the ball is on the ground, 
the height is zero, so we have h = 0.  Then -16(t-1)~2 + 212 = 0.

If we subtract 212 from both sides, we have -16(t-1)^2 = -212.
You can finish the problem by first diving both sides by -16, then 
taking the square root of both sides, and then adding one to both 
sides. 

This will give you the value of t that corresponds to h = 0, that 
is the time at which the height of the ball is 0.

The second problem seems harder.  We want to know when the ball is 
at the highest point.  One way to do this is to use calculus or 
geometry to find the maximum of the graph of the function h(t).  
This graph is a parabola so depending on what you know its not too 
bad.  If you know calculus, just find the maximum of the graph by 
setting the derivate of h(t) with respect to t equal to 0.  If you 
don't know calculus, you can still find the maximum by thinking 
about the equation.  We know that h(t) = -16(t-1)~2 + 212.

We can separate this into a positive part and a negative part.  
212 is always greater than zero no matter what t is, and 
-16(t-1)^2 isalways less than or equal to zero no matter what t 
is.  To get the biggest value of h we just find the value of t 
that gives us the smallest negative part.  You will find that a 
particular value of t gives us exactly zero for the negative part, 
so the maximum height will be 212.  

I hope that these hints are helpful.  Good luck

-Doctor Aaron,  The Math Forum