Completing the square problem? (Answer what you can please)

Question

You have 240 feet of wooden fencing to from two adjacent rectangular corrals. You want each corral to have an area of 1000 square feet. Draw a picture that represent the corral, write an equation for the perimeter of the corral in terms of length, write an equation for the area of the coral using length and width, and write an equation for the area in terms of length. Determine all possible dimensions of each coral by completing the square.

Answer 1

Sam,

Since the corrals are adjacent they share one wall, 

Let a = length of shared wall.  There are 3 of those if both corrals have the same length in one direction

Let b = length of corral 1 in other direction of the rectangle

Let c = length of corral 2 in other direction of the rectangle, in general b & c are NOT equal

Let P = the length of the fencing needed (given as 240 ft) which includes the shared wall a

Let A = summed area of both rectangles = 2000 sq. ft since each is 1000 sq. ft.

 

Then,

P = 3a + 2(b + c) = 240

A = a(b + c) = 2000 , solving for (b + c) = 2000/a because a is non-zero

P = 3a + 2(2000/a) = 240,  or multiplying both sides by a

3a² + 4000 = 240a ,  or 3a² -240a + 4000 = 0  or a² -80a + 4000/3 =0.  Now let's complete the square

(a - 40)² + (1333.33 - 1600) = 0    since we added 1600 to complete the square, we must also subtract it to maintain the original equality.

Simplifying:  (a - 40)² = 266.667

Solving by taking square roots of both sides:  (a-40) = +- 16.333  or a = 56.333 ft. and a = 23.667 ft.

And (b + c) = 2000/a  = 2000/56.333 = 35.50 ft   and 2000/23.667 = 84.51 ft. depending on a

And since each corral has an area of 1000 sq. ft., then ab = 1000 and ac = 1000 so you can also solve for b & c separately for each solution for a:

So b = 1000/56.333 = 17.75 ft = c;

Or b = 1000/23.667 = 42.25 ft. = c

 

Check:

P = 3(56.333) + 2(35.50) = 169.0 + 71.0 = 240.0 ft. for this solution pair: a = 56.333 & (b + c) = 35.50 ft

P = 3(23.667) + 2(84.51) = 71.0 + 169.0 = 240.0 ft. for the second solution pair: a = 23.667 & (b + c) = 84.51 ft.