The Garden Fence Problem
A gardener wants to fence a rectangular garden. She has 60 meters of fencing material. She wants the garden to have an area of at least 200 square meters. She also wants the length to be at least twice the width. Additionally, there is a tree in the garden that must be outside the fenced area; the tree is located 3 meters from one corner along the length and 2 meters from the same corner along the width. The gardener plans to put the fence such that the tree is outside. She can choose any rectangular dimensions as long as the perimeter is 60m and the area is at least 200 sq m. Determine possible dimensions that satisfy all constraints, and check if the tree will be outside the fence for those dimensions.
1. Let the length be L and width be W. Write the equations for perimeter and area, and express the constraints mathematically.
Read the worked answer
Perimeter: 2L + 2W = 60 => L + W = 30 => W = 30 - L. Area: A = L × W = L(30 - L) >= 200. Length at least twice width: L >= 2W => L >= 2(30 - L) => L >= 60 - 2L => 3L >= 60 => L >= 20. Also W > 0 => L < 30. So L must be between 20 and 30 (inclusive? L=20 gives W=10, area=200; L=30 gives W=0 invalid). So 20 <= L < 30. Area constraint: L(30 - L) >= 200 => 30L - L^2 >= 200 => L^2 - 30L + 200 <= 0 => (L-10)(L-20) <= 0 => 10 <= L <= 20. Combine with L >= 20: L must be exactly 20. Then W = 10. Check: L=20, W=10, perimeter=60, area=200, L=2W exactly. So the only dimensions satisfying all constraints are L=20m, W=10m.
2. Given the only possible dimensions (L=20, W=10), determine if the tree will be outside the fenced area. The tree is 3m from one corner along the length and 2m from the same corner along the width. Assume the corner is the origin (0,0), and the garden extends along positive x (length) and positive y (width). The tree is at (3,2). Is the tree inside or outside the rectangle?
Read the worked answer
The garden rectangle has corners at (0,0), (20,0), (20,10), (0,10). The tree at (3,2) has x=3 between 0 and 20, and y=2 between 0 and 10. Therefore, the tree is inside the rectangle. So with the only feasible dimensions, the tree is inside the fenced area, which violates the requirement that the tree must be outside. Thus, no dimensions satisfy all constraints simultaneously. The gardener must adjust either the fencing length, the area requirement, or the tree location.