(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 ...
Question 1 Report
(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.
Answer Details
(a) A circle inscribed in a square has diameter equal to the side, so side \(= 2r.\) Then \[\text{perimeter of square} + \text{circumference} = 4(2r) + 2\pi r = 100.\] \[8r + 2\times\frac{22}{7}r = 100 \Rightarrow 8r + \frac{44}{7}r = 100 \Rightarrow \frac{56r + 44r}{7} = 100.\] \[\frac{100r}{7} = 100 \Rightarrow r = \mathbf{7\,\text{cm}}.\]
(b) Perimeter of rectangle \(= 60\,\text{cm}\), with length \(= 4\times\)breadth. \[2(l + b) = 60 \Rightarrow l + b = 30,\quad l = 4b \Rightarrow 5b = 30 \Rightarrow b = 6.\]
(i) Length \(l = 4(6) = \mathbf{24\,\text{cm}}.\)
(ii) Diagonal \(= \sqrt{l^2 + b^2} = \sqrt{24^2 + 6^2} = \sqrt{576 + 36} = \sqrt{612} \approx \mathbf{24.7\,\text{cm}}.\)
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