Chris wants to make an enclosed rectangular area for a mulch pile. She wants to make the enclosure in such a way as to use a corner of her back yard. She also wants it to be twice as long as it is wide. Since the yard is already fenced, she simply needs to construct two sides of the mulch pile enclosure. She has only 15 feet of material available. Find the dimensions of the enclosure that will produce the maximum area
You imposed two conditions on the problem
1. the length is twice the width, and
2. the sum of length and width is 15
so clearly 2x + x = 15
x = 5
So it no longer is a problem dealing with maximimums.
the enclosure is 5 by 10 for an area of 50
(the maximum area would have been obtained by having two equal sides of 7.5 feet for an area of 56.25 feet^2, clearly larger than the 50 from above. But your condition of length = twice the width would not have been followed)
To find the dimensions of the enclosure that will produce the maximum area, we can use optimization techniques. Let's start by assigning variables to the dimensions of the enclosure.
Let's say the width of the enclosure is "x" feet. According to the given conditions, the length of the enclosure would be twice the width, so the length is "2x" feet.
We can calculate the perimeter using the available material:
Perimeter = 2(width) + length
15 = 2x + 2(2x)
15 = 2x + 4x
15 = 6x
Divide both sides by 6:
x = 2.5
So, the width of the enclosure is 2.5 feet, and the length is twice the width, which is 5 feet.
Now let's calculate the area of the enclosure:
Area = length × width
Area = 5 × 2.5
Area = 12.5 square feet
Therefore, the dimensions that will produce the maximum area for the enclosure are a width of 2.5 feet and a length of 5 feet, resulting in an area of 12.5 square feet.
To find the dimensions of the enclosure that will produce the maximum area, we need to maximize the area function. Let's assume the width of the enclosure is "w" feet.
Since Chris wants the enclosure to be twice as long as it is wide, the length of the enclosure would be 2w feet.
To calculate the area of the enclosed rectangular area, we multiply the length and the width:
Area = length * width
Area = (2w) * w
Area = 2w^2
We now have the area function in terms of the width, which is 2w^2.
Since Chris wants to use a corner of her backyard, only two sides of the rectangular area need to be constructed. Therefore, the total material required is the sum of the length and the two widths of the enclosure:
Total Material = length + width + width
Total Material = 2w + w + w
Total Material = 4w
From the given information, we know that Chris has only 15 feet of material available, so we can set up the equation:
4w = 15
Solving for w, we find:
w = 15 / 4
w ≈ 3.75 (rounded to two decimal places)
Since the width cannot be negative, we discard the negative solution and only consider the positive solution.
Therefore, the width of the enclosure is approximately 3.75 feet.
Now, to find the length, we multiply the width by 2:
Length = 2w
Length = 2 * 3.75
Length ≈ 7.5 feet
So, the dimensions of the enclosure that will produce the maximum area with 15 feet of material available are approximately 3.75 feet by 7.5 feet.