Herbal Garden Path Width Decision
The eco-club of a Class X section is designing a rectangular herbal garden 20 m long and 12 m wide inside the school compound. A uniform-width gravel path is to run along all four sides inside the boundary, and the remaining inner rectangle will be planted. The club has budget for gravel to cover exactly 64 m² of path. The teacher advises that the inner planting area must be at least 8 m by 5 m so that the medicinal plants have enough sun. A student sets up an equation and gets two possible widths, one of which is clearly impossible. The club must decide the width to the nearest 0.1 m and check whether the planting-area condition is met. The gardener also warns that a width below 1 m will be too narrow for wheelbarrows.
1. Let x be the path width. Form an equation for a path area of 64 m² and solve it.
Read the worked answer
Inner dimensions are (20−2x) by (12−2x). Path area = 20×12 − (20−2x)(12−2x) = 64. 240 − (240 − 64x + 4x²) = 64 64x − 4x² = 64 x² − 16x + 16 = 0. Using the quadratic formula: x = [16 ± sqrt(256−64)]/2 = [16 ± sqrt192]/2 = 8 ± 4sqrt3. So x ≈ 8 + 6.928 = 14.928 m or x ≈ 8 − 6.928 = 1.072 m.
2. Interpret both roots and decide the feasible width, including whether the planting-area and gardener conditions are met.
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Since the garden is only 12 m wide, 2x < 12, so x < 6 m. The root 14.928 m is impossible; only 1.072 m is feasible. Inner rectangle ≈ 17.856 m by 9.856 m, which is larger than 8 m by 5 m, so the planting condition is met. The feasible width is about 1.07 m, or 1.1 m if rounded to the nearest 0.1 m. The 1.07 m width is above the gardener's 1 m minimum.
3. The gardener insists on rounding up to 1.1 m. Calculate the extra gravel area and state whether the exact 64 m² budget is violated.
Read the worked answer
Using x = 1.1 m: inner dimensions = 20−2.2 = 17.8 m and 12−2.2 = 9.8 m. Inner area = 17.8×9.8 = 174.44 m². Path area = 240 − 174.44 = 65.56 m². Extra gravel area = 65.56 − 64 = 1.56 m². So rounding up to 1.1 m violates the exact budget by 1.56 m² of gravel; the club must either accept a small overspend or use the exact 1.07 m width.