A rectangular rug has an area of 18 square feet and a perimeter of 18 feet. What are the dimensions of the rug?
____feet by _____Feet
See question by Jr below this one.
L*W = 18 Ft.^2.
L = 18/W.
2L+2W = 18 Ft.
2*18/W + 2W = 18
36+2W^2 = 18W
W^2-9W+18 = 0. 18 = (-6)*(-3). sum = -6-3 = -9 = B.
(W-6)(W-3) = 0
W-6 = 0, W = 6.
W-3 = 0, W = 3.
Choose Width = 3 ft.
L = 18/W = 18/3 = 6 ft.
Let the length of the rectangular rug be 'l' and the width be 'w'.
From the given information, we can set up two equations:
1. Area of rectangle = length × width = 18 square feet.
Therefore, lw = 18.
2. Perimeter of rectangle = 2(length + width) = 18 feet.
Therefore, 2(l + w) = 18.
Simplifying the second equation:
2l + 2w = 18,
l + w = 9.
Now we have a system of two equations:
l × w = 18 (1)
l + w = 9 (2)
From equation (2), we can express l as l = 9 - w.
Substituting into equation (1):
(9 - w) × w = 18
9w - w^2 = 18
Rearranging:
w^2 - 9w + 18 = 0
Factoring the quadratic equation:
(w - 6)(w - 3) = 0
Solving for 'w':
w - 6 = 0 or w - 3 = 0
w = 6 or w = 3
If w = 6, then l = 9 - 6 = 3.
If w = 3, then l = 9 - 3 = 6.
Therefore, the possible dimensions of the rug are:
3 feet by 6 feet
or
6 feet by 3 feet.
To find the dimensions of the rectangular rug, we can use the information given about its area and perimeter.
1. Let's assume the length of the rug is L feet, and the width is W feet.
2. The formula for the area of a rectangle is A = length × width. Since we know the area is 18 square feet, we can write the equation as 18 = L × W.
3. The formula for the perimeter of a rectangle is P = 2(length + width). Since we know the perimeter is 18 feet, we can write the equation as 18 = 2(L + W).
4. We have two equations derived from the area and perimeter:
- Equation 1: 18 = L × W
- Equation 2: 18 = 2(L + W)
5. We can solve Equation 2 for either L or W and substitute it into Equation 1, then solve for the other variable.
- Let's solve Equation 2 for L: 18 = 2L + 2W. Rearranging this equation, we get 2L = 18 - 2W, or L = 9 - W.
6. Substitute L = 9 - W into Equation 1: 18 = (9 - W) × W.
7. Simplify Equation 1: 18 = 9W - W^2.
8. Rearrange Equation 1: W^2 - 9W + 18 = 0.
9. Factor Equation 1: (W - 3)(W - 6) = 0.
10. Solve for W: W = 3 or W = 6.
11. Substitute W = 3 and W = 6 back into Equation 2: L = 9 - W.
- When W = 3, L = 9 - 3 = 6.
- When W = 6, L = 9 - 6 = 3.
12. Therefore, the dimensions of the rectangular rug are:
- 6 feet by 3 feet
- or
- 3 feet by 6 feet