(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 . [T...

Answer Details

(a) Divide \(11111111_{two}\) by \(101_{two}\)

We use binary long division. Divisor is \(101_{two}\).

         1 1 0 0 1 1
       ___________________
1 0 1 ) 1 1 1 1 1 1 1 1
        1 0 1
        -----
          1 0 1
          1 0 1
          -----
            0 0 1 1
                1 1 1
                1 0 1
                -----
                  1 0 1
                  1 0 1
                  -----
                    0 0

The quotient is \(11111111_{two} \div 101_{two} = 110011_{two}\), remainder \(0\).

Check in base ten: \(11111111_{two}=255\), \(101_{two}=5\), and \(255 \div 5 = 51 = 110011_{two}\). Correct.

(b) Sector: radius 6 cm, angle 105°, \(\pi=\tfrac{22}{7}\)

(i) Perimeter. First find the arc length.

\[\text{Arc} = \frac{\theta}{360}\times 2\pi r = \frac{105}{360}\times 2\times\frac{22}{7}\times 6 = \frac{7}{24}\times\frac{264}{7} = 11\text{ cm}\]

Perimeter = arc + the two straight radii:

\[P = 11 + 2(6) = 23\text{ cm}\]

(ii) Area.

\[A = \frac{\theta}{360}\times \pi r^{2} = \frac{105}{360}\times\frac{22}{7}\times 6^{2} = \frac{7}{24}\times\frac{792}{7} = 33\text{ cm}^{2}\]