The area of a dining room is 160 square feet. a rectangular rug placed in the center of the room is twice as long as it is wide. If the rug is bordered by feet of hardwood floor on all sides, find the dimensions of the rug.
umm, how many feet?
To find the dimensions of the rug, we can set up a system of equations based on the given information.
Let's assume the width of the rug is x feet. Since the length of the rug is twice the width, the length would be 2x feet.
We know that the area of the dining room is 160 square feet. This means:
Width of dining room * Length of dining room = 160
To find the width of the dining room, we'll need to divide the area by the length:
Width of dining room = 160 / Length of dining room
The width of the dining room is also equal to the sum of the width of the rug and the border:
Width of dining room = Width of rug + 2 * Border width
Substituting 2x for the width of the rug, we get:
Width of dining room = 2x + 2 * Border width
Now we can set up the equation:
160 / Length of dining room = 2x + 2 * Border width
To solve for Length of dining room, we'll need to rearrange the equation:
Length of dining room = 160 / (2x + 2 * Border width)
Since the rug is bordered by hardwood floor on all sides, the dimensions of the rug would be:
Width of rug = x
Length of rug = 2x
So the dimensions of the rug would be x feet (width) by 2x feet (length), where x is a positive number.
To find the values of x and Border width, we'll need to substitute the equation for Length of dining room into the equation Width of dining room = 2x + 2 * Border width:
Width of dining room = 160 / (2x + 2 * Border width) = 2x + 2 * Border width
Now we have an equation with two variables, x and Border width. With this equation, we can solve for x and Border width by substituting x = 2 in the equation and then solving for Border width:
Width of dining room = 2(2) + 2 * Border width
160 / 6 = 2 + 2 * Border width
Simplifying further:
160 / 6 - 2 = 2 * Border width
26.67 - 2 = 2 * Border width
24.67 = 2 * Border width
Border width = 24.67 / 2 = 12.335 feet (approximately)
With the Border width, we can find the value of x:
Width of dining room = 2x + 2 * Border width
2(2) + 2 * 12.335 = 4 + 24.67 = 28.67 feet (approximately)
So, the dimensions of the rug would be:
Width of rug = x = 2 feet
Length of rug = 2x = 2(2) = 4 feet
Therefore, the dimensions of the rug would be 2 feet (width) by 4 feet (length).