word problem

Question

At a tennis club, a 16,500ft2 rectangular area is partitioned into three rectangular courts of equal size. A total of 860feet 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? Select all that apply.

Answer 1

OK...drawing a picture first always helps.  

 

So, what we have are 3 rectangular tennis courts, side-by-side, and all with the same dimensions.  Now, think of the larger rectangle that outlines all of them.  I can assign a variable y to the width of the large rectangle and the variable x to the width of each of the 3 smaller rectangles.  You'll note that y is the length of each of the smaller rectangles (make sure you draw a picture).  That means that the length of the larger rectangle is 3 times the width of the smaller rectangle, or 3x.

 

1. Now, we know that the area of the larger rectangle is 16500ft².  Then, looking at the dimensions of the larger rectangle in terms of x and y, we get:

 

3x * y = 16500 ft² (area of large rectangle)

 

2. We also know that the length of fence required to enclose the 3 courts (including interior fences) is 860 ft.

 

If you look at your picture, you'll see that there are 4 sides with length y and 6 sides with length x.

 

Then, the length of fence required for all sides is:

 

4y + 6x = 860 ft.

 

3.  Now, we have 2 equations that model the situation"

 

a. 3x * y = 16500 ft²

b. 4y + 6x = 860 ft

 

4. We can solve this system of equations since we have 2 equations and 2 variables.  How?

 

5. We'll use substitution:  Solve one of the equations for one of the variables and then substitute the result into the 2nd equation.  Here we go, ready?

 

Let's solve equation b for the variable y.

 

4y + 6x = 860 ft  => 4y = -6x +860 => y = -(6/4)x + 215 (which is 860/4).

 

Changing (6/4) to a decimal gives us:

 

c. y = -1.5x + 215

 

Since we now have y, substitute its value into equation a like this:

 

3x * (-1.5x +215) = 16500 ft²

 

Now, solve for x by expanding the left side of the equation above.

 

-4.5x² + 645x = 16500 ft²

 

6. We now have a quadratic equation to solve.  Without going through the solution of the equation, we'll find that

 

x = 110, 100/3  (must be two solutions for x since the equation is a quadratic)

 

So, are both of these solutions correct?  Let's plug these values of x into equation c and find out what y is.

 

c. y = -1.5x + 215, so for x = 110, y = -165 + 215 = 50

 

for x = 100/3, y = -1.5(100/3) + 215 => y = -(3/2)(100/3) + 215 => y = 50 + 215 = 265.

 

OK, we need to check each of these solutions to make sure they work.

 

Substitute the x,y pairs into either equation a or b.  Let's try a:

 

a. 3x * y = 16500 ft² => 3(110) * 50 = =16500

 

How about equation b?

 

b. 4y + 6x = 860 => 4(50) + 6(110) = 860

 

Good!  So, one possible dimension of the large rectangle is:

 

330 ft (100 x 3) by 265 ft.

 

What about the other x? 100/3?  I'll leave that to you!