Loading [MathJax]/jax/output/CommonHTML/fonts/TeX/fontdata.js

User interface language: English | Español

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 cmmin1.

Find the rate at which sand is being poured, in cm3min1, 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) (cm3min1)     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) (cm3min1)     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) (cm3min1)     A1

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 6 - Core: Calculus » 6.2 » Related rates of change.

View options