(a) Medi-Cap is a company that makes hollow spherical capsules for medicine.
One of its spherical capsules has a volume of 128π mm3.
Calculate the radius of this capsule. Give your answer in the form \(a\sqrt[3]{12}\), where \(a\) is an integer. You must show all your working. [4]
(b) Medi-Cap also makes another shape of capsule, called a Bullet capsule.
These capsules are in the shape of two hollow hemispheres attached to each end of a hollow cylinder, as shown below. The diameter of the cylinder, and of each hemisphere, is 8 mm.
Diagram not drawn to scale
To make a capsule easier to swallow, the exterior surface is coated with a glaze. Calculate the surface area that is covered by the glaze. Give your answer in terms of π in its simplest form. [3]
From the formula list: Volume of sphere \(= \dfrac{4}{3}\pi r^3\); Surface area of sphere \(= 4\pi r^2\)
Mark scheme (a)
Scheme
Marks
Comments
\(\dfrac{4}{3}\) × π × radius3 = 128π or equivalent
M1
If an equation is not seen, only award if appropriate calculations with 128, 4 and 3 seen Note: simplifying the cube root of 128 alone does not imply M1
(radius3 =) \(\dfrac{128\pi \times 3}{4 \times \pi}\) or equivalent
m1
radius3 = 96 OR (radius =) \(\sqrt[3]{96}\)
A1
(radius =) \(2\sqrt[3]{12}\) (mm)
B1
Must be from correct working FT ‘their derived 96’ provided their answer can be written the form \(a\sqrt[3]{12}\) An unsupported \(2\sqrt[3]{12}\) (mm) is awarded M0m0A0B0
Mark scheme (b)
Scheme
Marks
Comments
(Total surface area =) π × 8 × 12 + 2 × \(\dfrac{4 \times \pi \times 4^2}{2}\) or equivalent
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