Question

Gabi is working on another backyard project. She is building a fence to keep deer out of her garden. One side of her garden is against a wall, so she only needs to build three sides of the fence. She has 100 feet of fencing. If she needs to enclose a 600 square foot area, what should the length and width of her fence be?xy = 600 2x + y = 100 y = -2x x(-2x + 100) = x x = 600 x2 – 50x = -300 (x – 25)2 = 325 |x – 25| = 5

Answer 1

Answer:

• length: 14 feet , width: 43 feet, or

• length: 86 feet, width: 7 feet

Both solutions are valid.

Explanation:

1. First assumption is that the shape of the fence is rectangular.

2. Second, assum the length parallel to the wall measure y feet, so the other two lengths, y, together with x will add up 100 feet

• 2x + y = 100

3. The, the area of the fence will be:

• length × width = xy = 600

4. Now you have two equation with two variables which you can solveL

• Solve for y in the first equation: y = 100 - 2x

• Substitute the value of y into the second equation: x (100 - 2x) = 600

5. Solve the last equation by completing squares:

• Distributive property: 100x - 2x² = 600
• Divide both sides by - 1: 2x² - 100x = - 600
• Divide both sides by 2: x² - 50x = -300
• Add the sequare of the half of 50 to both sides: x² - 50x + 625 = 325
• Factor the left side: (x - 25)² = 325
• Square root both sides: x - 25 = ± 18.028

• Clear x: x = 25 ± 18.028
• x = 43.028 ≈ 43 or x = 6.972 ≈ 7

Both values are valid,

If x = 43 , then y = 600/43 = 14

If x = 7, then y = 600/7 = 86

Thus, the lenght and width of the fence may be:

• 43 feet (width) and 14 feet (length), or

• 7  feet (width) and 86 feet (length).