Theory question 111
Mensuration of Plane Shapes(a) In the diagram, ABCD is a rectangular garden (3n − 1)m long and (2n + 1) m wide. A wire-mesh 135 m long is used to mark its boundary and to divide it into 8 equal plots. Find the value of n.
(b) A cylinder with base radius 14 cm has the same volume as a cube of side 22 cm. Calculate the ratio of the total surface area of the cylinder to that of the cube. [Take π = 22/7]
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(a) The wire runs along 3 lines of length (3n − 1) (top, middle, bottom) and 5 lines of length (2n + 1) (two ends and three dividers).
3(3n − 1) + 5(2n + 1) = 135 → 19n + 2 = 135 → n = 7.
(b) 22/7 × 142 × h = 223 = 10648, so h = 10648/616 ≈ 17.29 cm.
Cylinder TSA = 2πr(r + h) = 2 × 22/7 × 14 × (14 + 17.29) ≈ 2753.1 cm2. Cube = 6 × 222 = 2904 cm2.
Ratio ≈ 2753.1 : 2904 = 73 : 77 (about 0.95 : 1).
Count every length of wire in the grid, including the inside dividers.