Finding dimensions of a rectangular area

Question

At a tennis club a 12825 ft square rectangular area is partitioned into three rectangular courts of equal size.  A total of 730 feet of fencing is used to enclose the three courts, including the interior sides. What are the possible dimensions in feet of the entire rectangular area?

Answer 1

Hi Gretchen,

This sounds like a problem of solving a system of equations.  How do I know that, you ask?  Why, because there are many unknowns - all those different possible dimensions - with many constraints.  So let's start trying to write some equations that represent the system.

 

I'm picturing three courts side-by-side-by-side (1-2-3), each court sharing its longer edge with its neighbor.  Around the entire set of three courts runs this rectangular fence, as well as fence along the longer edge between courts 1&2 and courts 2&3

 

Let w=the width of a single court, and h=the length of the court.  Then the width of all three courts together is 3*w. 

 

The amount of fencing, 730ft, can be equated to 730 = 2*(3*w + h) + 2*(h) .  The first term in parenthesis is  the outer perimeter, and the second term is the span of fence between the adjacent courts.

 

That equation has two unknowns, w and h.  So we need a second equation with the same set of unknowns.  Well we know that the entire area is 12825 = 3*w*h.

Here are the equations:

730 = 6*w + 4*h     (1st equation)

12825 = 3*w*h      (2nd equation)

 

Can this system of equations be solved?  YES!!  Arrange the first equation so that you have w = (4*h - 730)/6, and then you can sub this expression for w into the second equation.  When you do that, you'll have 12825 = 3*[(4*h-730)/6]*h. Notice that the part in square brackets is the expression for w, as determined from the first equation.

 

OK, the equation 12825 = 3*[(4*h - 730)/6]*h can be solved for h.  And when you have h, you can go back to either the 1st OR 2nd equation and determine w.  I'll leave this to you.

 

A few of points. 

1) Notice that the last equation I wrote, the one that you'll solve for h, is a quadratic equation.  You'll have to use the quadratic formula to solve it.  Hence, there may be two solutions to this problem (one h with its associated w, and another h with its associated w).

2) The problem asks for the dimensions of the entire rectangular area.  So you must state your answer as (h, 3*w), since the entire rectangular area is three times wider than any single court.

3) I suspect the geometry I've thought up here (side-by-side-by-side) is the one that your teacher/book is thinking of.  However, there may be different geometries that satisfy the problem - imagine two courts side-by-side, and another rotated 90Deg with its long side adjacent to the two short sides of the other courts.  If you include solutions to those in your assignment, you'll probably impress you teacher very much.  Just make sure to include descriptive drawings, so that your teacher can follow your thought process.

 

Hope this helps.

 

Best,

Paul