Question

Matthew constructs a fence around a patch of grass in his backyard. The patch has a width that is 8 feet more than 4 times the length. What is the perimeter of the fence if Matthew's patch of grass has an area of 5,472 square feet?

Answer 1

To find the perimeter of Matthew's patch of grass fence first determine the length by solving the quadratic equation derived from the area, then calculate the perimeter using the length and width.

Step-by-step explanation:

The problem describes a situation where Matthew constructs a fence around his backyard that has a specific relationship between its width and length and a given area. The width of the patch is '8 feet more than 4 times the length', which means if we denote the length by L, the width would be 4L + 8 feet. Given the area of the patch as 5,472 square feet, we can set up the area equation L × (4L + 8) = 5,472. To find the perimeter (P), we first need to resolve this equation to find the value of L, and then substitute back into P = 2L + 2W to get the total length of the fence required.

Let's solve for L:

L × (4L + 8) = 5,472
→ 4L² + 8L - 5,472 = 0

After solving this quadratic equation for L, we use the lengths L and W (where W = 4L + 8) to calculate the perimeter:

P = 2L + 2(4L + 8)

This arithmetic and algebraic approach is similar to understanding and comparing other geometrical figures and their dimensions, as exemplified in the reference questions provided.

Answer 2

P=380 ft.

Step-by-step explanation:

Area=length x width

Perimeter= 2*length + 2*width

Width= 4*length + 8

A=5472 ft²

5472= l * (4l +8) (plug in values)

5472 = 4l² +8l (simplify)

4l² + 8l - 5472 = 0 (subtract 5472)

4(l² + 2l -1368) = 0 (factor out a 4)

4(l +38)(l-36)=0 (factor)

l=36 ft. (-38 is extraneous because it is negative)

5472=38*w

w=152 ft.

P=2(38) + 2(152)

P= 380 ft.