You are planning a rectangular patio with length that is 7 ft less than three times its width. The area of the patio is 120 ft^2. What are the dimensions of the patio?
I just need to figure out what equation to write from this. I'm thinking something along the lines of 3x^2-7=120 (from (3x-7)(x)=120).
If it is, then I would solve it like this:
3x^2-7x=120
(3x^2-7x)/3=120/3
x^2-2.33x=40
x^2-2.33x+1.36=20+1.36
(x-1.16)^2=(plus or minus the square root of)41.16
x-1.16~(plus or minus)6.42
x-1.16~6.42 OR x-1.16~-6.42
x~7.58 OR x~-5.26
Am I right?
To find the dimensions of the patio, you can follow these steps:
1. Let's assume the width of the patio is x ft.
2. According to the problem, the length is 7 ft less than three times its width. So, the length would be (3x - 7) ft.
3. The area of a rectangle can be calculated by multiplying its length and width. In this case, we have the area as 120 ft^2.
4. Using the length and width values we found, we can create the equation for the area: x * (3x - 7) = 120.
5. Now, we can solve this equation to find the dimensions of the patio.
Going through your calculations, you made a small mistake when simplifying the equation after dividing both sides by 3:
3x^2 - 7x = 120
x^2 - (7/3)x = 40
When you subtracted 1.36 from both sides, you should have subtracted it from the right side instead of adding it:
x^2 - (7/3)x = 40 - 1.36
x^2 - (7/3)x = 38.64
Then, when you attempted to solve for x, you didn't correctly take the square root of both sides. The correct steps would be:
(x - 1.16)^2 = 38.64
Taking the square root of both sides:
x - 1.16 = ±√38.64
Now, let's calculate the values of x:
For x - 1.16 = +√38.64:
x = 1.16 + √38.64 ≈ 7.58
For x - 1.16 = -√38.64:
x = 1.16 - √38.64 ≈ -5.26 (Discard this negative value since dimensions cannot be negative)
Therefore, the width of the patio is approximately 7.58 ft. The length can be calculated using the equation we found earlier: 3x - 7.
Length = 3(7.58) - 7 ≈ 15.74 ft
Thus, the dimensions of the patio are approximately 7.58 ft by 15.74 ft.
Well, I doubt that -5.26 is a solution, but you can easily check that 7.58 works just fine.
Don't you check your answers when you get them?