Question 23

The maximum volume of a cylinder is cut from a cube of edge $$a$$. The volume of the remaining solid is $$ka^3$$, where $$k = \frac{p}{q}$$, $$\gcd(p, q) = 1$$. Taking $$\pi = \frac{22}{7}$$, the value of $$p + q$$ is

Correct Answer: 17

The largest cylinder has radius $$\dfrac{a}{2}$$ and height $$a$$, so its volume is

$$\pi \dfrac{a^3}{4} = \dfrac{22}{7} \times \dfrac{a^3}{4} = \dfrac{11a^3}{14}$$.

The remaining solid has volume

$$a^3 - \dfrac{11a^3}{14} = \dfrac{3a^3}{14}$$, so

$$k = \dfrac{3}{14}$$ and $$p + q = 3 + 14 = 17$$.

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