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A cylinder is to be cut out of the circular face of a solid hemisphere - AQA - A-Level Maths Pure - Question 9 - 2020 - Paper 2
Question 9
A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry. The cylinder has height $h$ a... show full transcript
Worked Solution & Example Answer:A cylinder is to be cut out of the circular face of a solid hemisphere - AQA - A-Level Maths Pure - Question 9 - 2020 - Paper 2
Step 1
Show that the volume, V, of the cylinder is given by
Answer
To find the volume of the cylinder, we use the formula for the volume of a cylinder:
pi r^2 h$$ where $r$ is the radius of the cylinder. From the diagram, we can see that the height of the cylinder $h$ and the radius of the hemisphere $R$ can be related through the equation: $$R^2 = r^2 + h^2$$ From this, we derive the radius of the cylinder: $$r = \\sqrt{R^2 - h^2}$$ Substituting this expression for $r$ back into the volume formula, we get: $$V = pi (R^2 - h^2) h = pi R^2 h - pi h^3$$ Thus, we have shown that the volume of the cylinder is correctly given by: $$V = pi R^2 h - pi h^3$$Step 2
Find the maximum volume of the cylinder in terms of R.
Answer
To find the maximum volume, we need to differentiate the volume with respect to height :
pi R^2 - 3\npi h^2$$ Setting the derivative equal to zero to find critical points for maximum volume: $$0 = pi R^2 - 3\npi h^2$$ This simplifies to: $$h^2 = \frac{R^2}{3}$$ Taking the positive root, we find: $$h = \frac{R}{\sqrt{3}}$$ Next, we must substitute this value of $h$ back into the volume formula to find the maximum volume: $$V = pi R^2 (\frac{R}{\sqrt{3}}) - pi (\frac{R}{\sqrt{3}})^3$$ Calculating this, we have: $$V = \frac{\npi R^3}{\sqrt{3}} - \frac{\npi R^3}{3\sqrt{3}}$$ Which further reduces to: $$V = \frac{2}{3} \npi \frac{R^3}{\sqrt{3}}$$ This is the maximum volume of the cylinder in terms of $R$.