# Solve Algebraic Question: Maximum Garden Area w/ 150ft Fence

• MM92
In summary, the question is asking for the dimensions of a rectangular garden with a 10 foot opening and a total of 150 feet of fencing material, in order to maximize the area. The answer is a rectangle with three sides measuring 40 feet and one side measuring 30 feet, resulting in a maximum area of 1600 square units. This can be found without using calculus, by comparing the area formula for a square and a rectangle with the same perimeter.
MM92
Hi everyone, I am new to this site. I was wondering if anyone here can help me answer a question, in order for me to study correctly for my math test tomorrow.

Here's the question:

Melissa plans to put a fence around her rectangular garden. She has 150 feet of fencing material to make the fence. If there is to be a 10 foot opening left for an entrance on one side of the garden, what dimensions should the garden be for maximum area? This question has to be answered in vertex form.

If anyone can answer this and explain it step by step to me that would be great!

You should fill in the details by answering the following:
(1)Fence and opening = ?
(2)What shape rectangle is maximum area for given perimeter?
(3)Using (1) and (2), what is length of side.?

mathman said:
You should fill in the details by answering the following:
(1)Fence and opening = ?
(2)What shape rectangle is maximum area for given perimeter?
(3)Using (1) and (2), what is length of side.?

but are you allowed to just know what shape of rectangle maximises area? or do you have to show it using calculus

anyway some clues

what is the total perimeter, given the 150 feet of fencing and the 10 feet of gap?

edit: whoops. i mean, ahem, is this number divisible by 4

Last edited:
Its late and i gtg, so ill post explanation tomoro, but it is, 3 sides are 40, and the side with the 10 gap is 30. getting you 1600 square units.

but are you allowed to just know what shape of rectangle maximises area? or do you have to show it using calculus

You don't need calculus. Area of square, side x is x2. Rectangle with same perimeter (not square) would have area (x+a)(x-a), which is obviously smaller.

mathman said:
You don't need calculus. Area of square, side x is x2. Rectangle with same perimeter (not square) would have area (x+a)(x-a), which is obviously smaller.
oh yeah, duh. you get used to general methods and forget the obv

## What is the maximum area of a garden with a 150ft fence?

The maximum area of a garden with a 150ft fence can be found by using the formula A = L * W, where A is the area, L is the length of the garden, and W is the width. In this case, the garden will have a rectangular shape, and the length and width must add up to 150ft. To find the maximum area, we can use the fact that the maximum area of a rectangle is when the length and width are equal. Therefore, the maximum area of the garden would be 75ft * 75ft = 5625 square feet.

## How can I find the dimensions of the garden with the maximum area?

To find the dimensions of the garden with the maximum area, we can use the same formula A = L * W and the fact that the length and width must add up to 150ft. We can set up an equation to represent this: A = (150 - W) * W. Then, we can use algebraic techniques (such as finding the derivative and setting it to zero) to find the value of W that will give us the maximum area. This value of W can then be used to find the length and width of the garden with the maximum area.

## Can the garden have any shape other than a rectangle?

No, the garden must have a rectangular shape in order to have the maximum area. This is because the formula for the area of a rectangle is A = L * W, which means that the area is only maximized when the length and width are equal. Any other shape would result in a smaller area.

## Is the maximum area the same for all gardens with a 150ft fence?

Yes, the maximum area will be the same for all gardens with a 150ft fence, as long as they have a rectangular shape. This is because the formula for the area of a rectangle (A = L * W) is the same for all rectangles, regardless of their dimensions. Therefore, the maximum area will always be 5625 square feet.

## How can I use this information to plan my garden?

This information can be used to plan your garden by knowing the maximum area that you can work with. You can use this information to determine the best dimensions for your garden and to make sure that you are using the space efficiently. Additionally, you can also use this information to budget for materials and plants, knowing the total area that your garden will cover.

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