A gardener has 720 feet of fencing to fence in a rectangular garden. One side of the garden is bordered by a river and so it does not need any fencing.What dimensions would guarantee that the garden has the greatest possible area?

Ah, the age-old question of creating the ultimate garden! Well, to maximize the area, you want to minimize the amount of fencing used, right? So let's put our clown hats on and try to solve this.

Since one side of the garden is already taken care of by the river, we only need to fence in three sides. Let's call the two sides we need to fence in "l" for length and the third side "w" for width. So, our garden perimeter equation is:

P = l + w + l

But we know that the total fencing available is 720 feet, so we can rewrite the equation as:

720 = l + w + l

Simplifying, we get:

720 = 2l + w

Now, let's solve for one variable in terms of the other. For example, let's isolate "l":

720 - w = 2l

Dividing both sides by 2, we get:

360 - w/2 = l

Now we have our area equation:

A = lw

Substituting our value for "l":

A = (360 - w/2)w

To find the maximum area, we need to determine the value of "w" that will give us the highest possible result. And guess what? This is where we bust out our joke:

Why did the scarecrow win an award? Because he was outstanding in his field!

Ahem, now back to our equation. We can use the quadratic function to find the maximum value of "w" that gives us the maximum area. But since I don't want to scare you away with quadratic formulas, I'll just tell you the answer:

To guarantee the greatest possible area, the garden should have dimensions of 180 feet by 180 feet.

To find the dimensions that would guarantee the garden has the greatest possible area, we need to maximize the area of the rectangular garden.

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

Since one side of the garden is bordered by a river and does not need fencing, the perimeter of the garden will be the sum of 3 sides only.

The perimeter of the garden is given as 720 feet. We can set up an equation based on the given information:

2L + W = 720

To express W in terms of L, we can isolate W:

W = 720 - 2L

Now, we can express the area (A) of the garden in terms of L and W:

A = L * W

Substituting the value of W from the previous equation:

A = L * (720 - 2L)

To find the dimensions that guarantee the greatest possible area, we need to find the maximum value of the area. The maximum area will occur at the vertex of a symmetric parabola, which is at the midpoint between its two x-intercepts.

The x-coordinate of the vertex can be found using the formula:

x = -b / (2a)

In our equation, a = -2 and b = 720, so substituting these values:

x = -720 / (2 * -2) = 720 / 4 = 180

Therefore, L = 180 feet.

Substituting this value of L back into the equation for W:

W = 720 - 2 * 180 = 720 - 360 = 360

Therefore, the dimensions that guarantee the greatest possible area of the garden are 180 feet by 360 feet.

To find the dimensions that would guarantee the garden has the greatest possible area, we can follow these steps:

Step 1: Understand the problem
We need to find the dimensions of the rectangular garden that maximize its area. We are given that one side of the garden is already bordered by a river and does not require any fencing. The total fencing available is 720 feet.

Step 2: Define the variables
Let's assume the width of the garden is "w" and the length is "l" (since the width can be any side of the rectangle).

Step 3: Formulate the equation(s)
We know that the perimeter of a rectangle can be expressed as: Perimeter = 2w + 2l.
In this case, since one side is adjacent to a river, the perimeter is equal to the sum of two widths and one length.
Therefore, we can write the equation as: 2w + l = 720.

Step 4: Simplify the equation(s)
We can rearrange the equation by isolating "l":
l = 720 - 2w.

Step 5: Find the area
The area of a rectangle is given by the formula: Area = length × width.
Substituting the equation for "l" into the area formula gives: Area = (720 - 2w) × w = 720w - 2w^2.

Step 6: Optimize the area
To maximize the area, we need to find the value of "w" that gives us the maximum value for the area equation defined in step 5. We can achieve this by finding the vertex of the parabolic equation 720w - 2w^2.

Step 7: Apply calculus

a. Take the derivative of the equation to find its critical points:
d(Area)/dw = 720 - 4w.

b. Set the derivative equal to zero and solve for "w" to find the critical points:
720 - 4w = 0.
4w = 720.
w = 180.

c. Check the nature of the critical point using the second derivative test:
d²(Area)/dw² = -4.
Since the second derivative is negative (-4), it means the critical point is a maximum.

Step 8: Solve for the length
Using the value of "w" from step 7 (w = 180), we can substitute it into our earlier equation to find the length:
l = 720 - 2w = 720 - 2(180) = 720 - 360 = 360.

Therefore, the dimensions that would guarantee the garden has the greatest possible area are width = 180 feet and length = 360 feet.

as always, divide the fencing so that it is evenly split between lengths and widths. You only have one length, because of the river. So, the maximum area is when the garden is 360ft x 180ft

extra credit: show that this is true.