(a) Divide (11111111_{two}) by (101_{two})
(b) A sector of radius 6 cm has an angle of 105° at the centre. Calculate its:
(i) perimeter ; (ii) area . [Take (pi = frac{22}{7})]
Explanation
(a) This can either be done using the long division method (easier) or convert everything to base ten and divide, then re-convert to base two.
I’ll be using the latter method.
(11111111_{two} = 1 times 2^{7} + 1 times 2^{6} + 1 times 2^{5} + 1 times 2^{4} + 1 times 2^{3} + 1 times 2^{2} + 1 times 2^{1} + 1 times 2^{0})
= (128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255_{ten})
(101_{two} = 1 times 2^{2} + 0 times 2^{1} + 1 times 2^{0} )
= (4 + 0 + 1 = 5_{ten})
(frac{255}{5} = 51_{ten})
| 2 | 51 |
| 2 | 25 r 1 |
| 2 | 12 r 1 |
| 2 | 6 r 0 |
| 2 | 3 r 0 |
| 2 | 1 r 1 |
| 0 r 1 |
= (110011_{two})
(b)
(i) Length of arc AXB = (frac{105}{360} times 2 times frac{22}{7} times 6cm)
= (11cm)
(therefore Perimeter = 6cm + 11cm + 6cm = 23cm)
(ii) Area of sector = (frac{105}{360} times frac{22}{7} times 6cm times 6cm)
= (33 cm^{2})