Height of the Ball

Date: 01/05/98 at 01:29:03
From: Lesley Trupiano
Subject: Algebra II Word Problem

Dear Dr. Math,

This question involves a "model," a concept that is hard for me to 
understand. 

The question is:

A football player kicked a 41-yard punt. The path of the ball was 
modeled by y= -0.035x(squared) + 1.4x + 1, where x and y are measured 
in yards. What was the maximum height of the ball? 

The player kicked the football toward midfield from the 18 yard line. 
Over which yard line was the ball when it was at its maximum height?

Basically I have problems with this because I know a) nothing about 
football whatsoever, and b) I don't understand how to tell where the 
maximum height of the ball is on my graph. I've plugged the equation 
into my calculator, but I still can't see how to do this.

Any help would be much appreciated!
Thanks,
Lesley

Date: 01/05/98 at 10:07:25
From: Doctor Rob
Subject: Re: Algebra II Word Problem

First of all, a model is a mathematical equation (or a set of several
mathematical equations) which describe some real, physical situation.
In this case, x measures the number of yards along the field from the
point where the ball was kicked, and y measures the number of yards
above the field the ball is.

One way to find the maximum is by completing the square. Write
y = -0.035*(x - a)^2 + b for some constants a and b. The maximum value 
of y will occur when the square is least. Since squares are always 
nonnegative, the square is least when it is zero. Then the maximum 
will occur when x = a, and the maximum value will be y = b. Since the 
ball was kicked from the 18-yard line, the answer to the second part 
will be 18 + a.

If you do not understand how to complete the square, write back again
and we will try to answer that.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/