(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...

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 \(= 73 : 77\)