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?

Question

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?

there are two rectangles sharing a boundary.

|Tomatoes | Beets |

Area for Tomatoes = 3x (Area for Beets)

Answer 1

If the vertical lines have length y, and the horizontal sides have length x, then

3y+2x = 270

and the total area is
a = xy = x(270-2x)/3 = 90x - 2/3 x^2

This is just a parabola with vertex at (135/2,6075/2), so the total area is 6075/2.

Since x=135/2, y=45

Since the inside rectangles have the same height (y), the tomatoes have width 135/2 * 3/4 and the beets have width 135/2 * 1/4