Date | May 2014 | Marks available | 5 | Reference code | 14M.2.hl.TZ2.9 |
Level | HL only | Paper | 2 | Time zone | TZ2 |
Command term | Find | Question number | 9 | Adapted from | N/A |
Question
Sand is being poured to form a cone of height h cm and base radius r cm. The height remains equal to the base radius at all times. The height of the cone is increasing at a rate of 0.5 cmmin−1.
Find the rate at which sand is being poured, in cm3min−1, when the height is 4 cm.
Markscheme
METHOD 1
volume of a cone is V=13πr2h
given h=r, V=13πh3 M1
dVdh=πh2 (A1)
when h=4, dVdt=π×42×0.5 (using dVdt=dVdh×dhdt) M1A1
dVdt=8π (=25.1) (cm3min−1) A1
METHOD 2
volume of a cone is V=13πr2h
given h=r, V=13πh3 M1
dVdt=13π×3h2×dhdt A1
when h=4, dVdt=π×42×0.5 M1A1
dVdt=8π (=25.1) (cm3min−1) A1
METHOD 3
V=13πr2h
dVdt=13π(2rhdrdt+r2dhdt) M1A1
Note: Award M1 for attempted implicit differentiation and A1 for each correct term on the RHS.
when h=4, r=4, dVdt=13π(2×4×4×0.5+42×0.5) M1A1
dVdt=8π (=25.1) (cm3min−1) A1
[5 marks]