Answer: Let x be the width of the carpet in feet.
Then, the length of the carpet is 3 feet more than three times the width, which can be expressed as 3x + 3.
The area of a rectangle is given by the formula A = length × width. Therefore, the area of the carpet is:
A = (3x + 3) x = 3x² + 3x
We know that the area of the carpet is 216 square feet, so we can set up the equation:
3x² + 3x = 216
To solve for x, we can first divide both sides by 3:
x² + x = 72
Then, we can rearrange the equation to get it in standard quadratic form:
x² + x - 72 = 0
Now we can use the quadratic formula to solve for x:
x = (-b ± √(b² - 4ac)) / 2a
where a = 1, b = 1, and c = -72.
Plugging these values into the formula, we get:
x = (-1 ± √(1² - 4(1)(-72))) / 2(1)
x = (-1 ± √(1 + 288)) / 2
x = (-1 ± √289) / 2
x = (-1 ± 17) / 2
This gives us two possible values for x: x = -9 or x = 8.
Since the width of the carpet can't be negative, we reject the negative solution and conclude that the width of the carpet is 8 feet.
To find the length of the carpet, we can use the expression we found earlier: 3x + 3. Plugging in x = 8, we get:
length = 3(8) + 3 = 24 + 3 = 27
Therefore, the dimensions of the carpet are 8 feet by 27 feet.
Explanation: