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.

To find the dimensions of the rectangle garden, we can start by understanding the given information. We know that we have only 3 sides of fencing, and one of the sides is a wall and does not require fencing.

Let's assume the length of the garden is L feet and the width is W feet.

Since the garden has only three sides of fencing (2 widths and 1 length), we know that the total length of fencing is L + 2W feet. Given that we have 16 feet of fencing available, we can write the equation as:

L + 2W = 16

Next, we are given that the area of the garden is 30 square feet. The area of a rectangle is calculated by multiplying the length and width. Thus, we can write another equation using the area:

L * W = 30

Now, we have a system of two equations with two unknowns. We can solve these equations simultaneously to find the values of L and W.

Using substitution method, we can solve the first equation for L:

L = 16 - 2W

Substituting this expression for L in the second equation, we get:

(16 - 2W) * W = 30

Simplifying this equation, we have:

16W - 2W^2 = 30

Rearranging the equation to put it in standard quadratic form:

2W^2 - 16W + 30 = 0

Now, we can solve this quadratic equation to find the values of W. We can either factorize it or use the quadratic formula:

Using the quadratic formula: W = (-b ± sqrt(b^2 - 4ac)) / 2a

For this equation, a = 2, b = -16, and c = 30.

Plugging in these values, we have:

W = (-(-16) ± sqrt((-16)^2 - 4 * 2 * 30)) / (2 * 2)

Simplifying further:

W = (16 ± sqrt(256 - 240)) / 4

W = (16 ± sqrt(16)) / 4

W = (16 ± 4) / 4

Simplifying the equation with both possibilities:

1. W = (16 + 4) / 4 = 20 / 4 = 5

2. W = (16 - 4) / 4 = 12 / 4 = 3

We obtained two values for W: 5 feet and 3 feet.

Now, we can substitute these values back into the equation L = 16 - 2W to find the corresponding values for L:

1. For W = 5:

L = 16 - 2 * 5 = 16 - 10 = 6

So, one possible set of dimensions for the rectangle garden is 6 feet for length (L) and 5 feet for width (W).

2. For W = 3:

L = 16 - 2 * 3 = 16 - 6 = 10

So, the other possible set of dimensions for the rectangle garden is 10 feet for length (L) and 3 feet for width (W).

Therefore, the dimensions of the rectangle garden, considering the given constraints, are either 6ft x 5ft or 10ft x 3ft.