Question
A company is designing a new logo in the shape of a letter “C”.
The letter “C” is formed between two circles with centre O. The point A lies on the circumference of the inner circle with radius \( r \) cm, where \( r < 10 \). The point B lies on the circumference of the outer circle with radius \( 10 \) cm. The reflex angle \( AOB \) is 5.2 radians. The letter “C” is shown by the shaded area in the following diagram.
(a) Show that the area of the “C” is given by \( 260 – 2.6r^2 \). [2]
(b) (i) Find the value of \( r \). [2]
(ii) Find the perimeter of the “C”. [3]
▶️Answer/Explanation
(a) The area of the “C” is the difference between the area of the outer sector and the inner sector: \[ \text{Area} = \frac{1}{2} \times 10^2 \times 5.2 – \frac{1}{2} \times r^2 \times 5.2 = 260 – 2.6r^2 \]
(b)(i) Given the area is 64 cm²: \[ 260 – 2.6r^2 = 64 \implies r^2 = \frac{196}{2.6} \implies r \approx 8.68 \, \text{cm} \]
(b)(ii) The perimeter is the sum of the outer arc, inner arc, and the two straight lines: \[ \text{Perimeter} = 10 \times 5.2 + 8.68 \times 5.2 + 2(10 – 8.68) \approx 99.8 \, \text{cm} \]