The diagram shows a wooden structure in the form of a cone, mounted on a hemispherical base. The vertical height of the cone is 24cm and the base height 7cm...
Question 1 Report
The diagram shows a wooden structure in the form of a cone, mounted on a hemispherical base. The vertical height of the cone is 24cm and the base height 7cm. Calculate, correct to three significant figures, the surface area of the structure. [Take \(\pi = \frac{22}{7}\)].
Answer Details
Surface area of the cone-on-hemisphere structure.
From the diagram, the common radius (from centre \(O\) to the base edge) is \(r=7\text{ cm}\), and the vertical height of the cone is \(h=24\text{ cm}\). The exposed surface is the curved surface of the cone plus the curved surface of the hemisphere; the flat circular join is internal and is not counted.
Slant height of the cone:
\[l=\sqrt{r^{2}+h^{2}}=\sqrt{7^{2}+24^{2}}=\sqrt{49+576}=\sqrt{625}=25\text{ cm}\]Curved surface area of the cone:
\[\pi r l=\frac{22}{7}\times7\times25=22\times25=550\text{ cm}^{2}\]Curved surface area of the hemisphere:
\[2\pi r^{2}=2\times\frac{22}{7}\times7^{2}=2\times\frac{22}{7}\times49=2\times22\times7=308\text{ cm}^{2}\]Total surface area:
\[550+308=858\text{ cm}^{2}\]Correct to three significant figures, the surface area is \(\mathbf{858\text{ cm}^{2}}\).
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