For a math homework assignment, Karla found the area and perimeter of a room of her house. She reported that the area of her rectangular living room is 180 square feet and that the perimeter is 54 feet. When drawing a sketch of her living room the next day, she realized that she had forgotten to write down the dimensions of the room. What are the dimensions of Karla's living room, in feet?
F. 9 by 20
G. 10 by 18
H. 12 by 15
J. 14 by 13
K. 16 by 11
w^2-27w+180=0
2w+2L=54
wL=180
w+L=27
180/w=15
180/w=12
(w-12) (w-15)
To find the dimensions of Karla's living room, let's represent the length of the room as "L" and the width of the room as "W". We can set up the following equations based on the given information:
1) The area of a rectangle is calculated by multiplying the length by the width: L * W = 180.
2) The perimeter of a rectangle is calculated by adding twice the length to twice the width: 2L + 2W = 54.
To solve this system of equations, we can use the substitution method.
From equation 1, we can isolate L by dividing both sides of the equation by W:
L = 180 / W.
Now, substitute this expression for L in equation 2:
2(180 / W) + 2W = 54.
Next, simplify the equation:
360 / W + 2W = 54.
To clear the fraction, multiply the entire equation by W:
360 + 2W² = 54W.
Rearrange the equation:
2W² - 54W + 360 = 0.
Let's try to factor this quadratic equation:
(W - 9)(2W - 40) = 0.
Setting each factor equal to zero, we get:
W - 9 = 0 or 2W - 40 = 0.
Solving each equation separately:
W = 9 or W = 20.
If W = 9, then L = 180 / 9 = 20.
If W = 20, then L = 180 / 20 = 9.
Therefore, the possible dimensions of Karla's living room are 9 by 20 (option F) or 20 by 9.
To find the dimensions of Karla's living room, we can use the formulas for area and perimeter of a rectangular shape.
The formula for the area of a rectangular shape is length multiplied by width, while the formula for the perimeter is the sum of all the sides.
Let's assign variables to the length and width of the living room. Let L represent the length, and W represent the width.
We are given that the area of the living room is 180 square feet. So, we have the equation:
L * W = 180
We are also given that the perimeter of the living room is 54 feet. The formula for the perimeter of a rectangle is 2L + 2W. So, we have the equation:
2L + 2W = 54
Now we have a system of two equations with two variables:
L * W = 180
2L + 2W = 54
To solve this system of equations, we can use substitution or elimination method, or even solve each equation for one variable and then substitute it into the other equation. However, in this case, we can use a different approach.
Let's consider the answer choices and test each one to see if it satisfies both equations:
F. 9 by 20:
If L = 9 and W = 20, then L * W = 9 * 20 = 180 (satisfies the area equation)
Also, 2L + 2W = 2 * 9 + 2 * 20 = 18 + 40 = 58 (does not satisfy the perimeter equation)
G. 10 by 18:
If L = 10 and W = 18, then L * W = 10 * 18 = 180 (satisfies the area equation)
Also, 2L + 2W = 2 * 10 + 2 * 18 = 20 + 36 = 56 (does not satisfy the perimeter equation)
H. 12 by 15:
If L = 12 and W = 15, then L * W = 12 * 15 = 180 (satisfies the area equation)
Also, 2L + 2W = 2 * 12 + 2 * 15 = 24 + 30 = 54 (satisfies the perimeter equation)
J. 14 by 13:
If L = 14 and W = 13, then L * W = 14 * 13 = 182 (does not satisfy the area equation)
K. 16 by 11:
If L = 16 and W = 11, then L * W = 16 * 11 = 176 (does not satisfy the area equation)
Based on the results, the only option that satisfies both equations is H. 12 by 15.
Therefore, the dimensions of Karla's living room are 12 feet by 15 feet.
P = 2L + 2W
You can find the dimensions by trial and error.
Since the perimeter ends in 4, A must be wrong because 18 times 2 is 36.