Rectangle garden using a wall as one side of the perimeter. What are the dimensions if we have 16ft of fencing and an area of 30sqft.
To be clear, you are only using the fencing for 3 sides of the garden.
well, Two of the sides would have to be either 5 or 3. Because 5+5+6=16 and 5x6=30 or 3+3+10=16 and 3x10=30.
Does this help?
No that does not help. Although that is probably the answer how does it get solved algebraically?
If the side of length x is parallel to the wall, then
x + 2y = 16
xy = 30
(16-2y)y = 30
2y^2 - 16y + 30 = 0
y^2-8y+15 = 0
(y-3)(y-5) = 0
y = 3 or 5
so, x = 10 or 6
The garden is 10x3 or 6x5
To find the dimensions of the rectangle garden, we can start by visualizing the problem. Let's assume that the wall is the longer side of the rectangle.
Let's denote the length of the wall as "L" and the width as "W". Since the wall is one side of the perimeter, we only need to calculate the lengths for the remaining three sides.
There are a few things we know:
1. The perimeter of a rectangle is the sum of all four sides.
2. We have 16 feet of fencing, which will be used for three sides of the rectangle.
3. The area of the rectangle is 30 square feet.
Now, let's proceed step by step to determine the dimensions of the rectangle:
1. Find the perimeter of the rectangle using the given 16 feet of fencing:
Perimeter = Length + Width + Length = 2 * Length + Width.
Since we have 16 feet of fencing, we can write:
2 * Length + Width = 16.
2. Express the width (W) in terms of the length (L) using the equation from step 1:
Width = 16 - 2 * Length.
3. Calculate the area of the rectangle using the length (L) and width (W):
Area = Length * Width = L(16 - 2L) = 30.
4. Simplify the equation to a quadratic equation:
16L - 2L^2 = 30.
5. Rearrange the equation to have one side equal to zero:
2L^2 - 16L + 30 = 0.
6. Solve the quadratic equation. You can use different methods such as factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
L = (-b ± sqrt(b^2 - 4ac)) / (2a).
Using the coefficients from the quadratic equation, a = 2, b = -16, and c = 30:
L = (-(-16) ± sqrt((-16)^2 - 4*2*30)) / (2*2).
Simplifying further, we get:
L = (16 ± sqrt(256 - 240)) / 4
= (16 ± sqrt(16)) / 4
= (16 ± 4) / 4.
Therefore, we have two possible values for L:
L1 = (16 + 4) / 4 = 20 / 4 = 5.
L2 = (16 - 4) / 4 = 12 / 4 = 3.
7. Substitute the values of L into the equation from step 1 to calculate the corresponding widths (W1 and W2):
For L1 = 5: Width = 16 - 2 * 5 = 6.
For L2 = 3: Width = 16 - 2 * 3 = 10.
So, we have two possible sets of dimensions for the rectangle garden:
1. Length (L1) = 5 ft and Width (W1) = 6 ft.
2. Length (L2) = 3 ft and Width (W2) = 10 ft.