3.13: Cone surface-area minimum; AP and GP

0606/23/M/J/21 — Question 10 · 15 marks

3.13 diagram
(a)
A cone has base radius xx, height yy and sloping edge x2+y2\sqrt{x^{2}+y^{2}}. Its volume is 10π10\pi.
Find yy in terms of xx and show that S=πxx2+900x4S=\pi x\sqrt{x^{2}+\dfrac{900}{x^{4}}}.
[3]
(b)
Find the exact value of xx for which SS is a minimum.
[5]
(a)(i)
An AP has first three terms 1p\dfrac{1}{p}, 1q\dfrac{1}{q}, −1q-\dfrac{1}{q}.
Show that d=−23pd=-\dfrac{2}{3p}.
[3]
(a)(ii)
The 1010th term is kp\dfrac{k}{p}. Find kk.
[2]
(b)
A GP has S∞=8S_{\infty}=8 and second term 32\dfrac{3}{2}. Find the two possible values of rr.
[5]