(a) A circle is inscribed in a square. If the sum of the perimeter of the square and the circumference of the circle is 100 cm, calculate the radius of the circle. [Take \(\pi = \frac{22}{7}\)].
(b) A rope 60cm long is made to form a rectangle. If the length is 4 times its breadth, calculate, correct to one decimal place, the :
(i) length ; (ii) diagonal of the rectangle.
Explanation
(a) Perimeter of a circle = \(2\pi r\).
Length of a side of square = 2r in terms of r.
Sum of perimeter of circle and square = \((2 \pi r) + (4 \times 2r)\)
\(2r (\pi + 4) = 100\)
\(r ( \frac{22}{7} + 4 ) = 50\)
\(r = 50 \div \frac{50}{7}\)
\(r = 7 cm\)
(b) (i) L = \(4 \times breadth\)
B = breadth.
60 cm = 2(4B + B)
2(5B) = 60 cm
\(B = \frac{60}{10} = 6 cm\)
\(L = 4 \times 6 = 24 cm\).
(ii) 
Let the diagonal be d.
\(d^{2} = 6^{2} + 24^{2}\)
\(d^{2} = 36 + 576\)
\(d^{2} = 612\)
\(d = \sqrt{612}\)
= \(24.74 cm \approxeq 24.7 cm\).