Hilda has a rectangular garden by the river in front of her house. The area she uses for planting tomatoes is three times as large as that she uses for beets. If she has 270 yards of fencing material to enclose the garden to protect it fro, rodents, will the total area attain a maximum, or, a minimum value? What would be the dimensions, and the value of the corresponding area?
of course the are will be minimum if the width of length is zero. But that's not what you want, I'm sure.
So, you want the arrangement of fences which will produce the maximum area.
You don't say how the area is to be divided. Is the beet garden enclosed by the tomatoes, or are the two separate enclosures, or are they two rectangles sharing a boundary? Do the beets sit in one corner of the tomato field? Is it just one large rectangle, with beets and tomatoes sharing the space? Each way gives a different answer.
Yes, there are two rectangles sharing a boundary.
|Tomatoes | Beets |
Thank you.
Area for Tomatoes = 3x (Area for Beets)
To find the dimensions and corresponding area that will give either a maximum or minimum value, we need to formulate the problem as a mathematical equation and solve it.
Let's denote the length of the rectangular garden as L and the width as W. Given that the area for planting tomatoes is three times that for beets, we can express the two areas as follows:
Area for tomatoes = 3x
Area for beets = x
The total area of the garden is the sum of these two areas:
Total area = 3x + x = 4x
Since we have 270 yards of fencing material, the perimeter of the garden is given by:
Perimeter = 2L + 2W = 270
We can rewrite this equation to solve for L:
L = 135 - W
Now, we can substitute this value of L into the equation for the total area:
Total area = 4x = (135 - W) * W
To find the maximum or minimum value, we need to take the derivative of the area equation and set it equal to zero:
d(Area)/dW = 0
Let's differentiate the equation with respect to W:
d(Area)/dW = 135 - 2W
Setting this equal to zero and solving for W:
135 - 2W = 0
2W = 135
W = 67.5
Now, let's substitute this value of W back into the equation for L:
L = 135 - W
L = 135 - 67.5
L = 67.5
So, the dimensions of the garden that will give either a maximum or minimum total area are W = 67.5 yards and L = 67.5 yards.
To find the corresponding area, we can substitute these values into the area equation:
Total area = (135 - W) * W
Total area = (135 - 67.5) * 67.5
Total area = 67.5 * 67.5
Total area = 4556.25 square yards
Therefore, the garden will attain a maximum total area of 4556.25 square yards with dimensions 67.5 yards by 67.5 yards.