(a) Find the smallest integer that satisfies the inequality \(x + 8 < 4x - 15\). (b) A sales girl is paid a monthly salary of N2,500 in addition to a commis...

Answer Details

(a) Smallest integer satisfying the inequality

\[x + 8 < 4x - 15\]

Collect like terms:

\[8 + 15 < 4x - x\]

\[23 < 3x\]

\[x > \frac{23}{3} = 7\tfrac{2}{3} \approx 7.67\]

The smallest integer greater than \(7.67\) is \(\mathbf{8}\).

(b) Monthly income of the sales girl

Fixed salary \(= \text{N}2{,}500\).

Commission \(= 5\text{ kobo in the naira} = \dfrac{5}{100}\text{ naira per naira} = \text{N}0.05\) per naira of sales.

Commission on N200,000 sales:

\[0.05 \times 200{,}000 = \text{N}10{,}000\]

Total income:

\[2{,}500 + 10{,}000 = \text{N}12{,}500\]

Her income for the month is \(\mathbf{\text{N}12{,}500.00}\).

(c) Area of the window

The window is a rectangle with a semicircle on top. From the diagram, the semicircular part has radius \(35\text{ cm}\), so the width of the rectangle equals the diameter:

\[\text{width} = 2 \times 35 = 70\text{ cm}, \qquad \text{height} = 50\text{ cm}\]

Area of rectangle:

\[A_1 = 70 \times 50 = 3{,}500\text{ cm}^2\]

Area of semicircle: (with \(\pi = \tfrac{22}{7}\))

\[A_2 = \frac{1}{2}\pi r^2 = \frac{1}{2}\times \frac{22}{7} \times 35^2 = \frac{1}{2}\times \frac{22}{7} \times 1225\]

\[A_2 = \frac{1}{2}\times 22 \times 175 = \frac{1}{2}\times 3{,}850 = 1{,}925\text{ cm}^2\]

Total area of window:

\[A = A_1 + A_2 = 3{,}500 + 1{,}925 = \mathbf{5{,}425\text{ cm}^2}\]