Cynthia Besch wants to buy a rug for a room that is 18 ft wide and 32 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 312 square feet of carpeting. What dimensions should the rug have?
If the uniform strip has width w, then we have
(18-2w)(32-2w) = 312
You can solve for w, or just note that the above gives you
(9-w)(16-w) = 78
Now, which two factors of 78 differ by 7?
(18 - 2x)(32 - 2x) = 312
4x^2 - 100 x + 576 = 312 ... solve for x ... the width of the strip of floor
the rug dimensions are the room dimensions minus the strip (on both sides)
To find the dimensions of the rug, we need to subtract the area of the room from the area of the rug, which leaves us with the area of the strip of floor.
Area of the room = length × width
Area of the room = 32 ft × 18 ft
Area of the room = 576 square ft
Area of the strip of floor = area of the rug - area of the room
Since the strip of floor is uniform, the dimensions of the room and the rug will differ by the same amount.
Let's assume the width of the strip of floor is x ft.
Area of the rug = [(width + 2x) × (length + 2x)]
Area of the rug = (18 ft + 2x) × (32 ft + 2x)
Given that the area of the rug is 312 square ft, we can set up the equation:
312 = (18 ft + 2x) × (32 ft + 2x)
Now we need to solve this equation to find the value of x and, consequently, the dimensions of the rug.
312 = (18 + 2x)(32 + 2x)
312 = 576 + 96x + 36x + 4x^2
0 = 4x^2 + 132x + 264 - 312
0 = 4x^2 + 132x - 48
To solve this quadratic equation, we can either factorize it or use the quadratic formula.
By factoring, we can start by dividing the equation by 4:
0 = x^2 + 33x - 12
Now we look for two numbers, let's call them a and b, that multiply to give -12 (the constant term) and add up to 33 (the coefficient of x). The numbers that satisfy this condition are 36 and -3.
So the equation becomes:
0 = (x + 36)(x - 3)
Setting each factor equal to zero:
x + 36 = 0 or x - 3 = 0
Solving for x:
x = -36 or x = 3
Since we are looking for the width of the strip of floor, x cannot be negative. So x = 3 ft.
Now we can find the dimensions of the rug:
Width of the rug = width of the room + 2 * width of the strip of floor
Width of the rug = 18 ft + 2 * 3 ft
Width of the rug = 18 ft + 6 ft
Width of the rug = 24 ft
Length of the rug = length of the room + 2 * width of the strip of floor
Length of the rug = 32 ft + 2 * 3 ft
Length of the rug = 32 ft + 6 ft
Length of the rug = 38 ft
So, the rug should have dimensions of 24 ft by 38 ft.
To find the dimensions of the rug, we need to subtract the extra strip of floor from the total dimensions of the room and calculate the area of the remaining space.
Let's assume the width of the strip of floor is x feet.
The width of the rug will be (18 - 2x) feet, as there are two strips of floor on either side.
The length of the rug will be (32 - 2x) feet, as there are two strips of floor on each end.
Now, we can calculate the area of the rug by multiplying its width and length:
Area of the rug = (18 - 2x) * (32 - 2x)
According to the given information, Cynthia can afford to buy 312 square feet of carpeting, so we can set up the equation:
(18 - 2x) * (32 - 2x) = 312
To solve this quadratic equation, we simplify and rearrange it:
576 - 50x + 4x^2 = 312
Rearranging again, we get:
4x^2 - 50x + 264 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use factoring as the method:
First, we divide the equation by 4 to simplify it:
x^2 - 12.5x + 66 = 0
Now, we factor the equation to find the values of x:
(x - 6)(x - 11) = 0
From this, we get two solutions:
x - 6 = 0 or x - 11 = 0
x = 6 or x = 11
Since we can't have a negative value for the width of the strip, we'll ignore the x = 11 solution.
Therefore, the width of the strip of floor is x = 6 feet.
Substituting this value back into our dimensions:
Width of the rug = 18 - 2(6) = 6 feet
Length of the rug = 32 - 2(6) = 20 feet
Therefore, the rug should be 6 feet wide and 20 feet long.