You are creating a garden whose length needs to be 5 feet less than its width and has an area of 84m^2
What should the dimensions be?
Let's assume the width of the garden is x feet.
According to the given information, the length needs to be 5 feet less than its width. So, the length would be x - 5 feet.
We are also given that the area of the garden is 84m^2.
To calculate the area, we need to convert the length and width into the same unit. As we are given the area in square meters, we should convert the dimensions from feet to meters.
1 feet is equal to 0.3048 meters. So, the width in meters would be x * 0.3048 and the length in meters would be (x - 5) * 0.3048.
The area in square meters is equal to the product of the length and width in meters. So, we have the equation:
(x * 0.3048) * ((x - 5) * 0.3048) = 84
Simplifying, we get:
0.09290304 * (x^2 - 5x) = 84
Multiplying both sides by 0.09290304:
x^2 - 5x = 84 / 0.09290304
x^2 - 5x = 902.2
Rearranging the equation, we get:
x^2 - 5x - 902.2 = 0
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Using the quadratic formula:
x = (-(-5) ± √((-5)^2 - 4(1)(-902.2))) / (2(1))
x = (5 ± √(25 + 3608.8)) / 2
x = (5 ± √(3633.8)) / 2
x = (5 ± 60.237) / 2
x ≈ (5 + 60.237) / 2 or x ≈ (5 - 60.237) / 2
x ≈ 32.6185 or x ≈ -27.6185
We can't have a negative width for our garden, so the width is approximately 32.6185 feet.
Substituting this value back into the equation for the length:
Length = 32.6185 - 5 = 27.6185 feet
Therefore, the dimensions of the garden should be approximately 32.6185 feet width and 27.6185 feet length.
To find the dimensions of the garden, we need to solve two equations. Let's say the width of the garden is w feet.
1) The length needs to be 5 feet less than the width:
Length = Width - 5
2) The area of the garden is 84m^2:
Area = Length * Width
84m^2 = (Width - 5) * Width
Now, we can solve the equation to find the value of Width:
84 = Width^2 - 5Width
To solve this quadratic equation, let's rearrange it:
Width^2 - 5Width - 84 = 0
Now, we can factorize or use the quadratic formula to find the value of Width. Let's use the quadratic formula:
Width = (-(-5) ± sqrt((-5)^2 - 4*1*(-84))) / (2*1)
Simplifying the equation further:
Width = (5 ± sqrt(25 + 336)) / 2
Width = (5 ± sqrt(361)) / 2
Width = (5 ± 19) / 2
So, we have two possible values for the width:
Width1 = (5 + 19) / 2 = 24 / 2 = 12 feet
Width2 = (5 - 19) / 2 = -14 / 2 = -7 feet (can't have a negative width)
Therefore, the width of the garden is 12 feet.
Now we can find the length using the first equation:
Length = Width - 5
Length = 12 - 5
Length = 7 feet
So, the dimensions of the garden should be 12 feet (width) by 7 feet (length).