When the glass contains water to a height $h$ cm, find the volume $V$ of water in terms of $h$ .

If the water in the glass evaporates at the rate of 3 cm³ per hour for each cm² of exposed surface area of the water, show that,

$\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3\sqrt {2\pi V}$ , where $t$ is measured in hours.

If the glass is filled completely, how long will it take for all the water to evaporate?

volume $= \pi \int_0^h {{x^2}{\text{d}}y}$     (M1)

$\pi \int_0^h {y{\text{d}}y}$     M1

$\pi \left[ {\frac{{{y^2}}}{2}} \right]_0^h = \frac{{\pi {h^2}}}{2}$     A1

[3 marks]

$\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3 \times$ surface area     A1

surface area $= \pi {x^2}$     (M1)

$= \pi h$     A1

$V = \frac{{\pi {h^2}}}{2} \Rightarrow h\sqrt {\frac{{2V}}{\pi }}$     M1A1

$\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3\pi \sqrt {\frac{{2V}}{\pi }}$     A1

$\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3\sqrt {2\pi V}$     AG

Note: Assuming that $\frac{{{\text{d}}h}}{{{\text{d}}t}} = - 3$ without justification gains no marks.

[6 marks]

${V_0} = 5000\pi$ ($= 15700$ cm³)     A1

$\frac{{{\text{d}}V}}{{{\text{d}}t}} =  - 3\sqrt {2\pi V}$

attempting to separate variables     M1

EITHER

$\int {\frac{{{\text{d}}V}}{{\sqrt V }}}  =  - 3\sqrt {2\pi } \int {{\text{d}}t}$     A1

$2\sqrt V  =  - 3\sqrt {2\pi t}  + c$     A1

$c = 2\sqrt {5000\pi }$     A1

$V = 0$     M1

$\Rightarrow t = \frac{2}{3}\sqrt {\frac{{5000\pi }}{{2\pi }}}  = 33\frac{1}{3}$ hours     A1

OR

$\int_{5000\pi }^0 {\frac{{{\text{d}}V}}{{\sqrt V }}}  =  - 3\sqrt {2\pi } \int_0^T {{\text{d}}t}$     M1A1A1

Note: Award M1 for attempt to use definite integrals, A1 for correct limits and A1 for correct integrands.

$\left[ {2\sqrt V } \right]_{5000\pi }^0 = 3\sqrt {2\pi } T$     A1

$T = \frac{2}{3}\sqrt {\frac{{5000\pi }}{{2\pi }}}  = 33\frac{1}{3}$ hours     A1

[7 marks]

This question was found to be challenging by many candidates and there were very few completely correct solutions. Many candidates did not seem able to find the volume of revolution when taken about the y-axis in (a). Candidates did not always recognize that part (b) did not involve related rates. Those candidates who attempted the question made some progress by separating the variables and integrating in (c) but very few were able to identify successfully the values necessary to find the correct answer.

This question was found to be challenging by many candidates and there were very few completely correct solutions. Many candidates did not seem able to find the volume of revolution when taken about the y-axis in (a). Candidates did not always recognize that part (b) did not involve related rates. Those candidates who attempted the question made some progress by separating the variables and integrating in (c) but very few were able to identify successfully the values necessary to find the correct answer.

This question was found to be challenging by many candidates and there were very few completely correct solutions. Many candidates did not seem able to find the volume of revolution when taken about the y-axis in (a). Candidates did not always recognize that part (b) did not involve related rates. Those candidates who attempted the question made some progress by separating the variables and integrating in (c) but very few were able to identify successfully the values necessary to find the correct answer.