4MA0 Higher January 2016 Paper 4 Q23
23 The diagram shows a solid cylinder.
Diagram NOT accurately drawn
The cylinder has radius \(4\sqrt{3}\) cm and height \(h\) cm.
The total surface area of the cylinder is \(56\pi\sqrt{6}\) cm²
Find the exact value of \(h\).
Give your answer in the form \(a\sqrt{2} + b\sqrt{3}\), where \(a\) and \(b\) are integers.
Show your working clearly.
(5)
Allow use of \(\pi\) = 3.14.. consistently throughout, for either scheme, but not a rounded decimal for the radius
| Scheme | Marks |
|---|---|
| \(\pi \times (4\sqrt{3})^2\) or \(2\pi \times (4\sqrt{3})^2\) or \(2\pi \times 4\sqrt{3}h\) | M1 |
| \(2\pi \times (4\sqrt{3})^2 + 2\pi \times 4\sqrt{3}h\) (\(= 56\pi\sqrt{6}\)) | M1 |
\(96\pi + 8\pi \times \sqrt{3}h\) (\(= 56\pi\sqrt{6}\)) (\(h\) =) \(\dfrac{56\pi\sqrt{6} - 2\pi(4\sqrt{3})^2}{2\pi(4\sqrt{3})}\) | M1 |
| (\(h\) =) \(\dfrac{56\pi\sqrt{6} - 96\pi}{8\pi\sqrt{3}}\) or (\(h\) =) \(\dfrac{56\sqrt{6} - 96}{8\sqrt{3}}\) oe | M1 |
| \(7\sqrt{2} - 4\sqrt{3}\) | A1 |
| (5) |
Notes
M1: correct expression for curved surface area or area of 1 or 2 circles (condone missing brackets around \(4\sqrt{3}\) for M1M1)
M1: correct expression for total surface area (candidate may have already divided by \(\pi\) or do so at any stage if already working in an equation)
M1: A correct simplified expression for total the total surface area or a correct expression for \(h\)
M1: correct rearrangement in terms of \(h\) and simplified as a quotient
A1: dep on M2
Alternative
| Scheme | Marks |
|---|---|
| \(2\pi rh + 2\pi r^2\) or \(2\pi rh + \pi r^2\) | M1 |
\(2rh + 2r^2 = 56\sqrt{6}\) or \(rh + r^2 = 28\sqrt{6}\) or \(2\pi r(h + r) = 56\pi\sqrt{6}\) or \(2\pi rh = 56\pi\sqrt{6} - 2\pi r^2\) | M1 |
| (\(h\) =) \(\dfrac{56\pi\sqrt{6}}{2\pi r} - r\) or (\(h\) =) \(\dfrac{56\pi\sqrt{6} - 2\pi r^2}{2\pi r}\) oe | M1 |
| (\(h\) =) \(\dfrac{56\pi\sqrt{6}}{2\pi 4\sqrt{3}} - 4\sqrt{3}\) or (\(h\) =) \(\dfrac{56\pi\sqrt{6} - 96\pi}{8\pi\sqrt{3}}\) oe | M1 |
| \(7\sqrt{2} - 4\sqrt{3}\) | A1 |
Notes
M1: A correct algebraic expression for the csa + 2 circles or csa + 1 circle
M1: A correct equation and correct division by \(\pi\) or \(2\pi\) or a correct factorisation or correctly isolating term in \(h\)
M1: A fully correct expression for \(h\)
M1: Correct substitution of \(r\) into a correct equation or expression for \(h\)
A1: dep on M2