The question is:

A gardener wishes to create a rectangular bed. There are 70 meters of fencing wire which will be used to fence three sides of the garden bed, with a brick wall forming the fourth side. Calculate the maximum area of the garden bed that can be created, and state it's dimensions.

Guys can you see where I went wrong.

I worked out this question by starting to work out the perimeter of the three sides.
P = 2x + y
70 = 2x + 7
y = 70 - 2x

Then I used it in the area equation:
A = lw
A = (70 - 2x)(x)

I got a max area of 612.5 m^2, which is correct.
And I got x = 0, x = 35. When I substitute 35 or 0 into the initial equations(70-2x) or (x). I get one of them as 0. Please help!

as you say, y = 70-2x, and the area

a = xy = x(70-2x)
You found the roots of that function which is where the area is zero!

The vertex of a parabola is midway between the roots, at x = 17.5
So, y=35 and the maximum area is indeed 612.5

Thanks for your answer! I released that I posted this twice on accident.

It seems like you made a mistake in your calculations. Let's go through the problem step by step to find the correct solution.

You correctly set up the equation for the perimeter:

P = 2x + y

Since there are three sides with fencing wire, the perimeter equation becomes:

P = 2x + y = 70

You correctly solved for y:

y = 70 - 2x

Now let's use the perimeter equation to rewrite the area equation:

A = x * y = x * (70 - 2x)

To find the maximum area, we need to find the maximum value of A. Let's differentiate the area equation with respect to x:

dA/dx = 70 - 4x

Set dA/dx to zero to find the critical point:

70 - 4x = 0

Now, solve for x:

4x = 70
x = 70/4
x = 17.5

So the value of x at the critical point is 17.5. Now substitute this value back into the perimeter equation to find y:

2x + y = 70
2 * 17.5 + y = 70
35 + y = 70
y = 70 - 35
y = 35

So the dimensions of the garden bed that maximize the area are x = 17.5 meters and y = 35 meters.

To find the maximum area, substitute these values back into the area equation:

A = x * y = 17.5 * 35 = 612.5 m^2

Therefore, the correct maximum area of the garden bed is 612.5 square meters, and its dimensions are 17.5 meters by 35 meters.

Let me walk you through the solution step by step.

To find the maximum area of the garden bed, we need to optimize the dimensions of the rectangle. We know that there are 70 meters of fencing wire available to fence three sides of the garden bed, and the fourth side will be a brick wall.

Let's start by defining our variables. Let x represent the length of the garden bed (in meters), and y represent the width of the garden bed (in meters).

We can use the information that the perimeter of the garden bed is formed by the fencing wire, so we have the equation:

Perimeter = 2x + y = 70 meters ----(1)

Since one side of the rectangle is a brick wall, we only need to fence the other three sides. The sum of the lengths of these sides should be equal to the total length of fencing wire, which is 70 meters.

Next, let's solve equation (1) for y in terms of x:

y = 70 - 2x ----(2)

Now, we move on to finding the area of the garden bed. The area A of a rectangle is given by the equation:

Area = length x width ----(3)

Substituting equation (2) into equation (3), we get:

A = x * (70 - 2x)

Now, we want to find the maximum value of A. To do this, we need to find the value of x that maximizes A. We can accomplish this by finding the vertex of the quadratic equation A = -2x^2 + 70x.

The x-coordinate of the vertex is given by x = -b/2a, where a = -2 and b = 70. Plugging in these values, we find:

x = -70 / (2 * -2) = 70 / 4 = 17.5 meters

Now that we have the value of x, we can substitute it back into equation (2) to find the value of y:

y = 70 - 2 * 17.5 = 70 - 35 = 35 meters

Therefore, the maximum area of the garden bed is:

A = x * y = 17.5 * 35 = 612.5 square meters

To summarize, the dimensions of the garden bed that will yield the maximum area are 17.5 meters by 35 meters, and the maximum area is 612.5 square meters.