Cynthia Besch wants to buy a rig for a room that is 18 ft wide and 25 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 120 square feet of carpeting. What dimensions should the rug have?
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To calculate the dimensions of the rug, we need to subtract the strip of floor from each side of the room. Let's represent the width of the strip of floor as 'x.'
So, the width of the rug will be (18 - 2x) and the length will be (25 - 2x).
The area of the rug can be calculated by multiplying the width and length:
Area of the rug = (18 - 2x) * (25 - 2x)
We know that Cynthia can afford 120 square feet of carpeting. So, we can set up the following equation:
(18 - 2x) * (25 - 2x) = 120
Now we can solve this equation to find the value of 'x' and then calculate the dimensions of the rug.
Let's simplify the equation:
(450 - 56x - 36x + 4x^2) = 120
Combine like terms:
4x^2 - 92x + 450 = 120
Rearrange the equation:
4x^2 - 92x + 450 - 120 = 0
Combine like terms:
4x^2 - 92x + 330 = 0
Divide the entire equation by 2 to simplify:
2x^2 - 46x + 165 = 0
Now we can either use factoring or the quadratic formula to solve for 'x'. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 2, b = -46, and c = 165.
Calculating the discriminant:
D = b^2 - 4ac
D = (-46)^2 - 4 * 2 * 165
D = 2116 - 1320
D = 796
Since the discriminant is positive, we will get two real solutions for 'x'.
Now we can substitute the values of 'a', 'b', and 'c' into the quadratic formula:
x = (-(-46) ± √(796)) / (2 * 2)
Simplifying the equation:
x = (46 ± √796) / 4
Calculating the square root of 796:
√796 ≈ 28.25
x = (46 ± 28.25) / 4
Now we can calculate the two possible values of 'x' by adding and subtracting 28.25 from 46:
x₁ = (46 + 28.25) / 4
x₁ = 74.25 / 4
x₁ ≈ 18.56
x₂ = (46 - 28.25) / 4
x₂ = 17.75 / 4
x₂ ≈ 4.44
Since the strip of floor should be a whole number, the width of the strip can be approximately 5 feet (x = 5).
Now we can calculate the dimensions of the rug:
Width = 18 - 2x
Width = 18 - 2 * 5
Width = 18 - 10
Width = 8 feet
Length = 25 - 2x
Length = 25 - 2 * 5
Length = 25 - 10
Length = 15 feet
Therefore, Cynthia Besch should buy a rug with dimensions 8 ft by 15 ft for the room.
To find the dimensions of the rug, we need to consider that it will have a uniform strip of floor around it. Let's assume the width of the strip is 'x' feet.
To find the dimensions of the rug, we need to subtract the width of the strip from the overall dimensions of the room. Therefore, the length and width of the rug will be (18 - 2x) and (25 - 2x) respectively.
The area of the rug can be calculated by multiplying the length and width of the rug:
Area of the rug = (18 - 2x) * (25 - 2x)
We are given that Cynthia can afford to buy 120 square feet of carpeting. So, we can set up the following equation:
(18 - 2x) * (25 - 2x) = 120
Now, solving this quadratic equation will give us the dimensions of the rug.
This equation may be simplified by expanding the terms:
(18 - 2x) * (25 - 2x) = 120
(450 - 36x - 50x + 4x^2) = 120
4x^2 - 86x + 450 - 120 = 0
4x^2 - 86x + 330 = 0
To solve this quadratic equation, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 4, b = -86, and c = 330. Plugging these values into the formula will give us the possible values of x.
Once we find the value of x, we can substitute it back into (18 - 2x) and (25 - 2x) to get the dimensions of the rug.
Which will leave a uniform strip all around?
10 by 12?
8 by 15?
6 by 20?