Decorative Object Composition

00:00 6 marks

A decorative object is made by joining a hemisphere to a cone. The radius of the hemisphere and the base of the cone is 5 cm. The height of the cone is 12 cm.

(a) Calculate the slant height of the cone. [2]
(b) Calculate the volume of the object. [2]
(c) Calculate the total surface area of the object (including the base of the hemisphere). [2]

Part a

3 minutes 2 marks

Question

Calculate the slant height of the cone. [2]

Hint Hint

Hint

Recall that the slant height l of a cone can be found using \(l^2 = h^2 + r^2\).

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Complete Answer

\(\sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\) cm

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Part b

3 minutes 2 marks

Question

Calculate the volume of the object. [2]

Hint Hint

Hint

Find the volume of the cone using \(V = \frac{1}{3}\pi r^2 h\) and the volume of the hemisphere using \(V = \frac{2}{3}\pi r^3\), then add them together.

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Complete Answer

\(V_{cone} = \frac{1}{3}\pi \times 5^2 \times 12 = 100\pi\; , \; V_{hemisphere} = \frac{2}{3}\pi \times 5^3 = \frac{250}{3}\pi\; \Rightarrow \; V = 100\pi + \frac{250}{3}\pi = \frac{550}{3}\pi \approx 576\) cm³

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Part c

4 minutes 2 marks

Question

Calculate the total surface area of the object (including the base of the hemisphere). [2]

Hint Hint

Hint

Determine the cone’s curved surface area, the hemisphere’s curved surface area, and add the area of the hemisphere’s circular base. Use \(A_{cone} = \pi r l\), \(A_{hemisphere (curved)} = 2\pi r^2\), and \(A_{base} = \pi r^2\).

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Complete Answer

Cone curved area: \(\pi \times 5 \times 13 = 65\pi\); Hemisphere curved area: \(2\pi \times 5^2 = 50\pi\); Base area: \(\pi \times 5^2 = 25\pi\). Total surface area = \(65\pi + 50\pi + 25\pi = 140\pi \approx 440\) cm².

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