Express h in terms of r.

Show that \(C = 20\pi {r^2} + \frac{{320\pi }}{r}\).

Given that there is a minimum value for C, find this minimum value in terms of \(\pi \).

correct equation for volume      (A1)
eg  \(\pi {r^2}h = 20\pi \)

\(h = \frac{{20}}{{{r^2}}}\)     A1 N2

[2 marks]

attempt to find formula for cost of parts      (M1)
eg  10 × two circles, 8 × curved side

correct expression for cost of two circles in terms of r (seen anywhere)      A1
eg  \(2\pi {r^2} \times 10\)

correct expression for cost of curved side (seen anywhere)      (A1)
eg  \(2\pi r \times h \times 8\)

correct expression for cost of curved side in terms of r      A1
eg  \(8 \times 2\pi r \times \frac{{20}}{{{r^2}}},\,\,\frac{{320\pi }}{{{r^2}}}\)

\(C = 20\pi {r^2} + \frac{{320\pi }}{r}\)      AG N0

[4 marks]

recognize \(C' = 0\) at minimum       (R1)
eg  \(C' = 0,\,\,\frac{{{\text{d}}C}}{{{\text{d}}r}} = 0\)

correct differentiation (may be seen in equation)

\(C' = 40\pi r - \frac{{320\pi }}{{{r^2}}}\)        A1A1

correct equation      A1
eg  \(40\pi r - \frac{{320\pi }}{{{r^2}}} = 0,\,\,40\pi r\frac{{320\pi }}{{{r^2}}}\)

correct working     (A1)
eg  \(40{r^3} = 320,\,\,{r^3} = 8\)

r = 2 (m)     A1

attempt to substitute their value of r into C
eg  \(20\pi  \times 4 + 320 \times \frac{\pi }{2}\)     (M1)

correct working
eg  \(80\pi  + 160\pi \)        (A1)

\(240\pi \) (cents)      A1 N3

Note: Do not accept 753.6, 753.98 or 754, even if 240\(\pi \) is seen.

[9 marks]