| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Topic | Connected Rates of Change |
| Type | Container filling: find depth rate |
| Difficulty | Standard +0.3 This is a standard connected rates of change problem requiring differentiation of the given volume formula with respect to time, then substituting known values. The volume formula is provided (no derivation needed), and it's a straightforward application of the chain rule dV/dt = (dV/dh)(dh/dt). Slightly easier than average due to the formula being given and requiring only one differentiation and algebraic rearrangement. |
| Spec | 1.02z Models in context: use functions in modelling1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08b Integrate x^n: where n != -1 and sums |
11.
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\caption{Figure 3}
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A bowl is modelled as a hemispherical shell as shown in Figure 3.\\
Initially the bowl is empty and water begins to flow into the bowl.\\
When the depth of the water is $h \mathrm {~cm}$, the volume of water, $V \mathrm {~cm} ^ { 3 }$, according to the model is given by
$$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$
The flow of water into the bowl is at a constant rate of $160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$ for $0 \leqslant h \leqslant 12$\\
Find the rate of change of the depth of the water, in $\mathrm { cm } \mathrm { s } ^ { - 1 }$, when $h = 10$\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q11 [5]}}