Height and volume of a cylinder with a given surface area
Height and volume of a cylinder with a given surface area
IP 2
10 marks
The total surface area of a closed cylinder of radius \(r\) cm and height \(h\) cm is \(S=2\pi r^2+2\pi rh\) square centimetres.
- Make \(h\) the subject of the formula.[2]
- A can has \(S=150\pi\) and \(r=5\). Find its height.[2]
- Show that the volume of a closed cylinder with total surface area \(S\) and radius \(r\) is \(V=\dfrac{rS}{2}-\pi r^3\). Use this to check the volume of the can in (b).[3]
- For \(S=150\pi\), show that \(250\pi-V=\pi(r-5)^2(r+10)\). Hence explain why no closed cylinder with this surface area holds more than the can in (b).[3]
Solution:
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Answer:(a) \(h=\dfrac{S-2\pi r^2}{2\pi r}\) (b) \(10\) cm (c) Shown; \(V=250\pi\text{ cm}^3\) both ways (d) \(250\pi-V=\pi(r-5)^2(r+10)\ge0\), equality only at \(r=5\)
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