the area of a rectangular rug is 24 square feet. The perimeter is 28 feet. What are the dimensions of the rug?
To find the dimensions of the rectangular rug, we can set up a system of equations based on the given information.
Let's assume the length of the rug is L and the width is W.
The area of a rectangle is calculated by multiplying its length by its width:
L * W = 24
The perimeter of a rectangle is calculated by adding the lengths of all its sides:
2L + 2W = 28
We now have a system of two equations:
Equation 1: L * W = 24
Equation 2: 2L + 2W = 28
To solve this system, we can use the substitution or elimination method. In this case, let's solve using substitution:
Solve Equation 1 for L or W:
L = 24 / W
Substitute this expression for L in Equation 2:
2(24 / W) + 2W = 28
Simplify the equation:
48 / W + 2W = 28
Multiply through by the common denominator to clear fractions:
48 + 2W^2 = 28W
Rearrange the equation to get a quadratic equation:
2W^2 - 28W + 48 = 0
We can solve this quadratic equation. Factoring becomes a bit tricky, so let's use the quadratic formula:
W = (-b ± sqrt(b^2 - 4ac)) / 2a
Plugging in the coefficients of the quadratic equation into the formula:
W = (-(-28) ± sqrt((-28)^2 - 4*(2)*(48))) / 2*(2)
Simplifying:
W = (28 ± sqrt(784 - 384)) / 4
W = (28 ± sqrt(400)) / 4
W = (28 ± 20) / 4
Using both possible solutions:
W1 = (28 + 20) / 4 = 48 / 4 = 12
W2 = (28 - 20) / 4 = 8 / 4 = 2
Since the width can't be 2 (as it would result in an area of 48 square feet), the width is 12 feet.
Substitute this value back into Equation 1:
L * 12 = 24
L = 24 / 12 = 2
Therefore, the dimensions of the rug are 2 feet by 12 feet.
you don't have choice ? 4 ft and 6ft
Let's look at the factors of 24
2 * 12
3 * 8
4 * 6
Now plug those numbers into this formula.
P = 2L + 2W
Which of those factors fits for the perimeter?