(a) In the diagram, ABCD is a rectangular garden (3n - 1)m long and (2n + 1)m wide. A wire mesh 135m long is used to mark its boundary and to divide it into...
Question 1 Report
(a)
In the diagram, ABCD is a rectangular garden (3n - 1)m long and (2n + 1)m wide. A wire mesh 135m 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 \(\pi = \frac{22}{7}\)]
Answer Details
(a) The diagram shows the rectangle divided into 4 columns and 2 rows, giving \(4 \times 2 = 8\) equal plots. Length \(AB = (3n-1)\ \text{m}\), width \(BC = (2n+1)\ \text{m}\).
The wire mesh forms the boundary and the internal dividing lines.
Horizontal lines (each of length \(3n-1\)): top, bottom and 1 internal line \(= 3\) lines.
Vertical lines (each of length \(2n+1\)): left, right and 3 internal lines \(= 5\) lines.
Total length of wire:
\[3(3n-1) + 5(2n+1) = 135\]\[9n - 3 + 10n + 5 = 135\]\[19n + 2 = 135\]\[19n = 133 \Rightarrow n = 7\]\(n = 7\)
(b) Take \(\pi = \frac{22}{7}\). Let the cylinder have radius \(r = 14\ \text{cm}\) and height \(h\).
Volume of cube (side \(22\ \text{cm}\)):
\[V = 22^3 = 10648\ \text{cm}^3\]Volume of cylinder equals this:
\[\pi r^2 h = \frac{22}{7} \times 14^2 \times h = 616h\]\[616h = 10648 \Rightarrow h = \frac{10648}{616} = \frac{121}{7}\ \text{cm} \;\left(=17\tfrac{2}{7}\right)\]Total surface area of cylinder:
\[2\pi r(r + h) = 2 \times \frac{22}{7} \times 14 \times \left(14 + \frac{121}{7}\right)\]\[= 88 \times \frac{98 + 121}{7} = 88 \times \frac{219}{7} = \frac{19272}{7}\ \text{cm}^2\]Total surface area of cube:
\[6 \times 22^2 = 6 \times 484 = 2904\ \text{cm}^2\]Ratio (cylinder : cube):
\[\frac{19272}{7} : 2904 = 19272 : 20328\]Dividing both by \(264\):
\[= 73 : 77\]Ratio \(= 73 : 77\)
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