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?
Tomatoes | Beets
Horizontal ______ ______
Vertical ______ ______
Area ______ _____
To determine whether the total area will attain a maximum or minimum value, we need to analyze the given information. Let's break down the problem step by step.
Let x be the width of the area planted with beets, and y be the length of the area planted with beets (in yards). Since the garden is rectangular, the length of the area planted with tomatoes will be y as well.
Based on the information provided, the area used for planting tomatoes is three times larger than the area used for planting beets. Therefore, the area of the tomato section would be 3xy, and the area of the beet section would be just xy.
The perimeter of a rectangle is given by the formula:
Perimeter = 2(length + width)
In this case, we are given that the total fencing material Hilda has is 270 yards. Therefore, we can write the equation as:
2(x + y + x + 3y) = 270
Simplifying the equation:
2(4x + 4y) = 270
8x + 8y = 270
4x + 4y = 135
To proceed, let's solve one equation for one variable. We'll isolate x by rewriting the equation as:
4x = 135 - 4y
x = (135 - 4y)/4
x = (135/4) - y
Now, we can substitute the value of x into the area equation for beets:
Beet area = xy = [(135/4) - y]y
Next, we need to find the derivative of the beet area equation with respect to y to determine whether the total area attains a maximum or minimum value. Differentiating the equation gives:
Beet area' = [(135/4) - y] - 2y
Setting the derivative equal to zero to find critical points:
[(135/4) - y] - 2y = 0
(135/4) - 3y = 0
(135/4) = 3y
y = (135/4) / 3
y = 45/4
Substituting this value of y back into the equation for x:
x = (135/4) - (45/4)
x = 90/4
x = 45/2
So, the dimensions of the garden for maximum or minimum area are x = 45/2 yards and y = 45/4 yards.
Finally, we can calculate the values of the corresponding areas:
Tomato area = 3xy = 3(45/2)(45/4) = (3/2)(45^2)/4 = 3037.5 square yards
Beet area = xy = (45/2)(45/4) = (45^2)/8 = 253.125 square yards
Therefore, the total area of the garden, including the tomato and beet sections, would be:
Total area = Tomato area + Beet area = 3037.5 + 253.125 = 3290.625 square yards.
To summarize, the garden area will attain a maximum or minimum value. The dimensions of the garden will be x = 45/2 yards and y = 45/4 yards, with a corresponding total area of 3290.625 square yards.