Rescaling a Garden Drawing When the Path and Margins Will Not Fit
A seventh grader is making a scale drawing of a rectangular community garden that is 6 m by 4.5 m. The chosen scale is 1 cm : 1.5 m, so the garden measures 4 cm by 3 cm. The student must draw a 0.75 m-wide path around all four sides. On the drawing, the path is 0.5 cm wide. The paper is 5 cm by 4 cm, and the teacher requires at least 0.2 cm of blank margin on every side. The student says, 'The drawing barely fits, so I'll just skip the margin.' The teacher responds that the margin is a constraint, not a suggestion. The student then suggests changing only the paper size, but the class has only the 5 cm by 4 cm paper. The student must keep the garden and path proportional and label the scale.
1. Calculate the actual outside dimensions of the garden plus path. Then determine whether the student's current scale can fit on the paper with the required margins. Show your reasoning.
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The path is 0.75 m wide on all four sides, so it adds 0.75 m twice to each dimension. Outside dimensions: length 6 + 0.75 + 0.75 = 7.5 m; width 4.5 + 0.75 + 0.75 = 6.0 m. At 1 cm : 1.5 m, the scale factor is 1/1.5 = 2/3 cm per meter. The outer drawing is 7.5 × (2/3) = 5 cm by 6.0 × (2/3) = 4 cm. The paper is 5 cm by 4 cm, and the teacher requires 0.2 cm margin on every side. Usable width is 5 - 0.2 - 0.2 = 4.6 cm; usable height is 4 - 0.2 - 0.2 = 3.6 cm. The 5 cm by 4 cm drawing exceeds both usable dimensions. Even though it exactly matches the paper's outer edge, it leaves no margin. Therefore the current scale cannot satisfy all constraints; skipping the margin is not a valid fix because the margin is part of the task.
2. Propose a new scale that fits all constraints and keeps the drawing as large as possible. Justify with calculations.
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Let s be the scale in cm per meter. The outer drawing dimensions are 7.5s and 6.0s. The usable paper is 4.6 cm by 3.6 cm. So 7.5s ≤ 4.6 and 6.0s ≤ 3.6. The first gives s ≤ 4.6/7.5 ≈ 0.613; the second gives s ≤ 3.6/6.0 = 0.600. The tighter constraint is s ≤ 0.600 cm/m. The largest exact scale is 0.6 cm per meter, which is equivalent to 1 cm : 1/0.6 m = 1 cm : 1.666... m, or 1 cm : 1.67 m. At that scale the outer drawing is 7.5×0.6 = 4.5 cm by 6.0×0.6 = 3.6 cm. With 0.2 cm margins, the required paper is 4.5 + 0.4 = 4.9 cm by 3.6 + 0.4 = 4.0 cm, which exactly fits the 5 cm by 4 cm paper. If the class requires a conventional '1 cm : n m' scale with a simple number, 1 cm : 2 m (s = 0.5 cm/m) also fits and leaves extra margin, but it is not as large as possible. The best answer states the exact maximum and then, if needed, a practical rounded scale.