| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Related rates with cones, hemispheres, and bowls (variable depth) |
| Difficulty | Standard +0.3 This is a standard related rates problem requiring differentiation of the given volume formula with respect to time, then substituting known values. It's slightly easier than average because the volume formula is provided (no geometry derivation needed) and it's a straightforward single-step application of the chain rule with basic algebraic manipulation. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
7.
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\caption{Figure 1}
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Figure 1 shows a hemispherical bowl.\\
Water is flowing into the bowl at a constant rate of $180 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$.\\
When the height of the water is $h \mathrm {~cm}$, the volume of water $V \mathrm {~cm} ^ { 3 }$ is given by
$$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 90 - h ) , \quad 0 \leqslant h \leqslant 30$$
Find the rate of change of the height of the water, in $\mathrm { cm } \mathrm { s } ^ { - 1 }$, when $h = 15$ Give your answer to 2 significant figures.
\hfill \mbox{\textit{Edexcel C34 2018 Q7 [5]}}