How do I set this up and solve?
You are planning a rectangular patio with length that is 7 ft less than 3 times its width. The area of the patio is 120 ft squared. What are the dimensions of the patio?
w(3w-7) = 120
3w^2 - 7w - 120 = 0
No rational solution.
x = (7+√1489)/6
To set up and solve this problem, we'll use the information given and set up an equation.
Let's assume the width of the patio is "w" feet. Since the length is 7 feet less than 3 times the width, the length would be (3w - 7) feet.
The area of a rectangle is calculated by multiplying the length by the width, so the equation to represent the area of the patio is:
Area = Length × Width
120 ft² = (3w - 7) ft × w ft
Now, we can solve this equation to find the value of "w" (the width), which will allow us to determine the dimensions of the patio.
To solve the equation, let's first distribute the terms:
120 ft² = 3w² - 7w
Next, rearrange the equation so that it is in standard quadratic form:
3w² - 7w - 120 ft² = 0
At this point, we have a quadratic equation that can be factored, completed by the square, or solved using the quadratic formula. Since the factors of "w" may not be readily apparent, I will use the quadratic formula:
w = (-b ± √(b² - 4ac)) / 2a
In this case, a = 3, b = -7, and c = -120. Plug in these values into the formula and simplify:
w = (-(-7) ± √((-7)² - 4(3)(-120))) / (2(3))
w = (7 ± √(49 + 1440)) / 6
w = (7 ± √1489) / 6
Now, we have two possible values for "w". To find the dimensions of the patio, we need to consider both options.
1. w = (7 + √1489) / 6
Substitute this value of "w" into the length equation:
Length = 3w - 7
Length = 3((7 + √1489) / 6) - 7
2. w = (7 - √1489) / 6
Substitute this value of "w" into the length equation:
Length = 3w - 7
Length = 3((7 - √1489) / 6) - 7
By calculating both options for length and width, you can determine the dimensions of the rectangular patio.