Cynthia besch wants to buy a rug for a room that is 19 ft wide and 27 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 345 Square feet of carpeting. What dimensions should the rug have?
If there is a strip of width x all around the rug, then
(19-2x)(27-2x) = 345
find x, then you can get the dimensions of the rug from the equation
Well, Cynthia certainly wants to make her room look un-"rugg-ed," huh? Let's do some carpet calculations together, shall we?
To find out the dimensions of the rug, we need to subtract the uniform strip of floor from the total room dimensions. Let's assume the width of the strip is represented by 'x'.
So, the length of the rug would be (27 - 2x) feet, considering there's a strip of floor on either side. Similarly, the width would be (19 - 2x) feet.
Now, we need to find the area of the rug, and since area is length multiplied by width, we can use the equation:
Area = (27 - 2x)(19 - 2x)
And since Cynthia can afford only 345 square feet of carpeting, we have:
(27 - 2x)(19 - 2x) = 345
Now, this equation might not look appealing, but don't worry! We'll solve it using math instead of magic tricks.
Simplifying the equation, we get:
4x^2 - 92x + 432 = 0
Using the quadratic formula, we find the value of 'x' to be approximately 5 feet.
So, the width of the rug would be (19 - 2x) = (19 - 2 * 5) = 9 feet,
And the length of the rug would be (27 - 2x) = (27 - 2 * 5) = 17 feet.
Voila! The rug should have dimensions of approximately 9 feet by 17 feet, giving Cynthia that perfect blend of coziness and spaciousness in her room.
Remember, though, if the laughter carpets your room, make sure you don't slip on it!
To find the dimensions of the rug, we need to subtract the uniform strip of floor from the overall dimensions of the room.
Let's assume that the uniform strip of floor that Cynthia wants to leave around the rug has a width of "x" ft.
The width of the rug would be the width of the room minus twice the width of the uniform strip of floor:
Width of rug = Width of room - 2 * Width of uniform strip
Similarly, the length of the rug would be the length of the room minus twice the width of the uniform strip of floor:
Length of rug = Length of room - 2 * Width of uniform strip
Now, let's substitute the given values into the equations:
Width of rug = 19 ft - 2x
Length of rug = 27 ft - 2x
The area of the rug is given as 345 square feet:
Area of rug = Width of rug * Length of rug
Substituting the expressions for width and length, we have:
345 = (19 - 2x) * (27 - 2x)
Simplifying further:
345 = 513 - 38x - 54x + 4x^2
0 = 4x^2 - 92x + 168 (rearranging the terms)
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use factoring:
0 = 4x^2 - 92x + 168
0 = (2x - 6)(2x - 28)
Now we have two possible values for "x":
2x - 6 = 0 OR 2x - 28 = 0
Solving the equations:
2x - 6 = 0
2x = 6
x = 3
2x - 28 = 0
2x = 28
x = 14
Since the uniform strip of floor can't be larger than the dimensions of the room, we discard the solution where x = 14.
Therefore, the width of the uniform strip of floor (and also the rug) is x = 3 ft.
Now, we can find the dimensions of the rug:
Width of rug = 19 ft - 2 * 3 ft = 19 ft - 6 ft = 13 ft
Length of rug = 27 ft - 2 * 3 ft = 27 ft - 6 ft = 21 ft
So, the rug should have dimensions 13 ft x 21 ft.
To find the dimensions of the rug, we need to subtract the area of the strip around the rug from the total area of the room.
Given:
Width of the room = 19 ft
Length of the room = 27 ft
Total area of the room = Width × Length
Total area of the room = 19 ft × 27 ft = 513 sq. ft
Now, let's assume the width of the strip around the rug as 'x' feet.
To calculate the area of the rug, we need to subtract the area of the strip from the total area of the room.
Area of the rug = Total area of the room - Area of the strip
Since the strip goes around all four sides of the rug, the width and length of the rug will be reduced by twice the width of the strip.
Width of the rug = (19 ft - 2x)
Length of the rug = (27 ft - 2x)
To calculate the area of the rug, use the formula:
Area of the rug = Width of the rug × Length of the rug
Substituting the values:
Area of the rug = (19 ft - 2x) × (27 ft - 2x)
The area of the rug should not exceed 345 sq. ft, so:
(19 ft - 2x) × (27 ft - 2x) ≤ 345 sq. ft
Solve this quadratic inequality to find the possible dimensions for the rug. The resulting dimensions should be positive and feasible for the given constraints.