(a) The diagram shows a wooden structure in the form of a cone, mounted on a hemispherical base. The vertical height of the cone is 48 m and the base radius...

Answer Details

(a) Reading the diagram. The structure is a cone (apex \(L\), base diameter \(MN\), centre \(O\)) sitting on a hemisphere of the same base radius. The cone has vertical height \(h = 48\text{ m}\) and base radius \(r = 14\text{ m}\). The exposed surface is the curved surface of the cone plus the curved surface of the hemisphere; the flat circular join is inside the solid and is not counted.

Step 1: Slant height of the cone.

\[l = \sqrt{r^{2}+h^{2}} = \sqrt{14^{2}+48^{2}} = \sqrt{196+2304} = \sqrt{2500} = 50\text{ m}.\]

Step 2: Curved surface area of the cone \(= \pi r l\):

\[= \frac{22}{7}\times 14\times 50 = 22\times 2\times 50 = 2200\text{ m}^2.\]

Step 3: Curved surface area of the hemisphere \(= 2\pi r^{2}\):

\[= 2\times\frac{22}{7}\times 14^{2} = 2\times\frac{22}{7}\times 196 = 2\times 22\times 28 = 1232\text{ m}^2.\]

Step 4: Total surface area of the structure.

\[= 2200 + 1232 = 3432\text{ m}^2 \approx 3430\text{ m}^2\ (3\text{ s.f.}).\]

(b) Ages problem. Let Musah's present age be \(M\) and Sesay's present age be \(S\).

Sum of present ages:

\[M + S = 100.\]

Five years ago Musah was twice as old as Sesay:

\[M - 5 = 2(S - 5) \;\Rightarrow\; M - 5 = 2S - 10 \;\Rightarrow\; M = 2S - 5.\]

Substitute into the sum:

\[(2S - 5) + S = 100 \;\Rightarrow\; 3S = 105 \;\Rightarrow\; S = 35.\]

Answers: (a) surface area \(= 3430\text{ m}^2\) (3 s.f.); (b) Sesay's present age is \(\mathbf{35}\) years.