Photo AI
A solid right circular cylinder has radius r cm and height h cm - Edexcel - A-Level Maths Pure - Question 1 - 2008 - Paper 2
Question 1
A solid right circular cylinder has radius r cm and height h cm. The total surface area of the cylinder is 800 cm². (a) Show that the volume, V cm³, of the cylinde... show full transcript
Worked Solution & Example Answer:A solid right circular cylinder has radius r cm and height h cm - Edexcel - A-Level Maths Pure - Question 1 - 2008 - Paper 2
Step 1
Show that the volume, V cm³, of the cylinder is given by
Answer
To derive the volume of the cylinder in terms of its radius r, we start with the formula for the total surface area of a cylinder, which is given by:
Solving for h gives:
The volume V of the cylinder is defined as:
Substituting the expression for h into the volume formula:
This matches the given equation, confirming that the volume is:
Step 2
use calculus to find the maximum value of V, to the nearest cm³.
Answer
To find the maximum volume, we first differentiate V with respect to r:
Setting the derivative equal to zero to find critical points:
\Rightarrow 3\pi r^2 = 400 \\ \Rightarrow r^2 = \frac{400}{3\pi} \\ \Rightarrow r = \sqrt{\frac{400}{3\pi}}$$ Calculating the value gives approximately: $$r \approx 6.5 ext{ cm (2 d.p.)}$$ Substituting this back into the volume formula to find V: $$V = 400 \left(\sqrt{\frac{400}{3\pi}}\right) - \pi \left(\sqrt{\frac{400}{3\pi}}\right)^2$$ Calculating this volume yields: $$V \approx 1737.25\text{ cm}^3$$ Thus, the maximum volume to the nearest cm³ is 1737 cm³.Step 3
Justify that the value of V you have found is a maximum.
Answer
To ascertain whether the critical point found is indeed a maximum, we will analyze the second derivative of V:
Since r is positive, (\frac{d^2V}{dr^2} < 0), indicating that the function is concave down at the critical point. This confirms that the value of V found at this radius is a maximum.