## Question 8:

**Summary method:**

| Answer | Mark | Guidance |
|--------|------|----------|
| Express $V$ in terms of $h$ | B1 | Correct substitution |
| Differentiate $V$ with respect to $h$ | M1 | NOT if $h=50$ or $r=50\tan30$ used |
| Attempt chain rule | M1 | Resulting equation must involve exactly 2 variables |
| Attempt separate variables | M1 | Their equation must involve exactly 2 variables |
| Correct integrals | A1 | Ignore limits |
| Substitute correct limits | M1 | Integrals must be of correct forms |
| Answer | A1 | |

**Example method 1:**

| Answer | Mark | Guidance |
|--------|------|----------|
| $V=\frac{\pi}{3}(h\tan30°)^2 h$ or $V=\frac{\pi}{3}\left(\frac{h}{\sqrt{3}}\right)^2 h$ oe | B1 | or $V=\frac{\pi}{9}h^3$ oe |
| $\frac{dV}{dh}=\frac{\pi}{3}h^2$ | M1 | Attempt differentiate their $V$ in terms of $h$ only. NOT if $h=50$ or $r=50\tan30$ used |
| $\frac{dV}{dt}=\frac{\pi}{3}h^2\frac{dh}{dt}$ oe or $\frac{dh}{dt}=\frac{3}{\pi h^2}\frac{dV}{dt}$ | M1 | Attempt use chain rule for $\frac{dV}{dt}$ or $\frac{dh}{dt}$ in terms of $t$ and $h$ only. (Set their $\frac{dV}{dt}=-2h$) |
| $``\frac{\pi}{3}h^2\frac{dh}{dt}"=-2h$ oe or $\frac{dh}{dt}=\frac{-6}{\pi h}$ | M1 | Attempt separate variables in terms of $h$ and $t$ only. Integral signs not essential |
| $\pi\int_{50}^{0} h\,dh = -\int_{0}^{t} 6\,dt$ oe | | |
| $\left[\frac{\pi h^2}{2}\right]_{50}^{0} = \left[-6t\right]_0^t$ oe | A1 | Correct integrals, any limits or none |
| $-\pi\times\frac{50^2}{2}=-6t$ | M1 | Substitute correct limits into integrals of forms $ah^2$ and $bt$ |
| Time $=\frac{625\pi}{3}$ secs or 654 secs (3 sf) oe | A1 | Allow without secs, or 10.9 mins or 10 mins 54 secs |

**Example method 2:**

| Answer | Mark | Guidance |
|--------|------|----------|
| $V=\frac{\pi}{3}r^2\cdot\frac{r}{\tan30°}$ or $V=\frac{\pi}{\sqrt{3}}r^3$ oe | B1 | Subst $h=\frac{r}{\tan30°}$ into correct formula for $V$ |
| $\frac{dV}{dr}=\sqrt{3}\pi r^2$ | M1 | |
| $\frac{dV}{dt}=\sqrt{3}\pi r^2\frac{dr}{dt}$ oe | M1 | Attempt chain rule to find $\frac{dV}{dt}$ or $\frac{dr}{dt}$ in terms of $t$ and $r$. (Set their $\frac{dV}{dt}=-2r\sqrt{3}$ oe) |
| $\pi\int_{\frac{50}{\sqrt{3}}}^{0} r\,dr = -\int_{0}^{t} 2\,dt$ oe | M1 | Attempt separate variables in terms of $r$ and $t$ only. Integral signs not essential |
| $\left[\frac{\pi r^2}{2}\right]_{\frac{50}{\sqrt{3}}}^{0} = \left[-2t\right]_0^t$ oe | A1 | Correct integrals, any limits or none |
| $-\frac{\pi\times50^2}{6}=-2t$ | M1 | Substitute correct limits into integrals of forms $ar^2$ and $bt$ |
| Time $=\frac{625\pi}{3}$ secs or 654 secs (3 sf) oe | A1 | Allow without secs, or 10.9 mins or 10 mins 54 secs |

## Question 8 (continued):

**Example Method 3 (NOT using chain rule)**

$V = \frac{\pi}{3}(h\tan 30°)^2 h$ or $V = \frac{\pi}{3}\left(\frac{h}{\sqrt{3}}\right)^2 h$ | **B1** | or $V = \frac{\pi}{9}h^3$

$h = \sqrt[3]{\frac{9V}{\pi}}$ | **M1** | Allow $h = kV^{1/3}$

$\frac{dV}{dt} = -2 \times \sqrt[3]{\frac{9V}{\pi}}$ | **M1** | $\frac{dV}{dt} = -2\times$(their $h$ in terms of $V$)

$\sqrt[3]{\frac{\pi}{9}}\int_{\frac{\pi 50^3}{9}}^{0} V^{-1/3}dV = -2[t]_0^t$ | **M1** | Attempt separate variables in terms of $V$ and $t$ only. Integral signs not essential

$\sqrt[3]{\frac{\pi}{9}} \times \frac{3}{2}\left[V^{2/3}\right]_{\frac{\pi 50^3}{9}}^{0} = -2t$ | **A1** | Correct integrals, any limits or none

$-\sqrt[3]{\frac{\pi}{9}} \times \frac{3}{2} \times \left(\frac{\pi 50^3}{9}\right)^{2/3} = -2t$ | **M1** | Substitute correct limits into integrals of forms $aV^{2/3}$ & $bt$; OR substitute $t=0$ & $V = \frac{\pi 50^3}{9}$ to find $c$ and substitute $V=0$

Time $= \frac{625\pi}{3}$ secs or 654 secs (3 sf) | **A1** | Allow without secs; or 10.9 mins or 10 mins 54 secs. SC: Use of $r = h\sin30$ (answer $\frac{625\pi}{4}$ or 491) can score all 4 M-marks and final A1

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