A homeowner wants to enclose a rectangular garden with fencing The garden will be adjacent to his neighbours lot. There will be fencing on all four sides. His neighbour will be paying for half of the shared fence.

What should the dimensions of the garden be if the area is 432 m^2 and the homeowner would like to keep his share of the cost to a minimum.

Good day, I'm currently stuck on my last homework question and have failed three times to answer it properly... I have worked with questions like this before but I was asked to maximize the area when given the total length of fencing used but I'm confused as to where to start when given the area...

Thank you very much for your time, it's very appreciated.

If the dimensions are x and y, then

xy = 432, so y = 432/x

If the shared side is y, then the cost for our homeowner is

c = 2x + y + y/2 = 2x + 3/2 y = 2x + 648/x

we want minimum cost, so

dc/dx = 2 - 648/x^2
dc/dx = 0 when x = 18

So, the garden is 18x24, with the 24' side shared.

Thank you very much Steve.

Good day! I'll be happy to help you with your homework question.

To enclose a rectangular garden, we need to consider the lengths of the four sides of the garden.

Let's assume the length of the garden is L (in meters) and the width is W (in meters).

From the given information, we know that the area of the garden is 432 m^2. Therefore, we have the equation:

L * W = 432

Now, let's consider the four sides of the garden:
- Side 1: L (adjacent to the neighbor's lot)
- Side 2: W
- Side 3: L (adjacent to the neighbor's lot)
- Side 4: W

Since the neighbor will pay for half of the shared fence, we want to minimize the cost for the homeowner. This means we want to minimize the total length of fencing used.

The total length of the fencing used is:
Side 1 + Side 2 + Side 3 + Side 4

To minimize the cost, we minimize the total fencing used. Therefore, we want to minimize the sum of all sides:

Side 1 + Side 2 + Side 3 + Side 4 = L + W + L + W = 2L + 2W

To minimize the total fencing, we need to minimize the equation 2L + 2W.

Now, we have two equations:
Equation 1: L * W = 432
Equation 2: 2L + 2W = ?

To minimize the total length of fencing used, we need to solve Equation 1 for one variable (either L or W) and substitute it into Equation 2.

Let's solve Equation 1 for L:
L = 432 / W

Substituting this into Equation 2, we get:
2(432 / W) + 2W = ?

To further simplify this equation, let's multiply each term by W:
2(432) + 2W^2 = ?

Simplifying further:
864 + 2W^2 = ?

To find the minimum value of the total fencing used, we can take the derivative of this equation with respect to W, set it equal to zero, and solve for W.

Let's differentiate the equation:
d(864 + 2W^2) / dW = 2(2W) = 4W

Setting the derivative equal to zero and solving for W:
4W = 0
W = 0

Since the width cannot be zero, this means there is no minimum value for the total length of fencing used. Therefore, there is no specific dimension for the garden that will minimize the homeowner's share of the cost.

I hope this explanation helps you understand the problem and the solution process. Let me know if you have any further questions!

Good day! I'm here to help you out with your homework question. Don't worry, I'll guide you through the process step by step so you can arrive at the correct answer.

To minimize the cost, we need to determine the dimensions of the garden that will result in the least amount of fencing required while still achieving an area of 432 m^2.

Let's break down the problem into smaller steps:

Step 1: Define the variables
Let's assume the length of the garden is L and the width is W. The shared fence will be on one side of the garden, so it will have a length of L. The other three sides will have a combined length of 2W.

Step 2: Determine the total length of the fencing
Based on the given information, we know that the neighbor will pay for half of the shared fence. So, the total length of the fencing can be calculated as:
Total fencing length = L + 2W + 0.5L

Step 3: Express one variable in terms of the other
We know that the area of the garden is 432 m^2, which can be expressed as:
Area = Length * Width
432 = L * W

From this equation, we can express one variable in terms of the other:
L = 432 / W

Step 4: Substitute the expression for one variable into the total fencing length equation
By substituting L = 432 / W into the total fencing length equation, we get:
Total fencing length = (432 / W) + 2W + 0.5(432 / W)

Step 5: Find the minimum value of the total fencing length
To find the minimum value, we can take the derivative of the total fencing length equation with respect to W, set it equal to zero, and solve for W.

Step 6: Solve for W
Differentiating the total fencing length equation, we get:
d/dW (Total fencing length) = -432/W^2 + 2 - 0.5(432/W^2)

Setting the derivative equal to zero, we have:
-432/W^2 + 2 - 0.5(432/W^2) = 0

Simplifying this equation and solving for W will give us the width of the garden.

Step 7: Calculate the length based on the width
Once we have the value of W, we can substitute it back into L = 432 / W to find the length of the garden.

Step 8: Check the dimensions
Finally, make sure to check if the dimensions satisfy the condition that the area is 432 m^2.

By following these steps, you should be able to find the dimensions of the garden that minimize the cost while achieving the desired area. Good luck with your homework!