Standard +0.3 This is a straightforward related rates problem requiring the chain rule (dV/dt = dV/dh × dh/dt) with one differentiation of a composite function. The setup is clear, the algebra is routine, and it's a standard C3 application question—slightly easier than average due to its direct structure and single-step reasoning.
3 The volume, \(V\) cubic metres, of water in a reservoir is given by
$$V = 3 ( 2 + \sqrt { h } ) ^ { 6 } - 192 ,$$
where \(h\) metres is the depth of the water. Water is flowing into the reservoir at a constant rate of 150 cubic metres per hour. Find the rate at which the depth of water is increasing at the instant when the depth is 1.4 metres.
Any non-zero constants \(k\), \(n\); condone presence of \(-192\) here
Obtain \(9h^{-\frac{1}{2}}(2+\sqrt{h})^5\) or unsimplified equiv
A1
Without \(-192\) now
Divide 150 by their derivative, algebraic or numerical
\*M1
Using any recognisable attempt at first derivative
Substitute \(h=1.4\) and evaluate
M1
Dep \*M; assume appropriate substitution if calculation goes wrong
Obtain \(0.06\) or \(0.060\) or \(0.0603\)
A1
But not greater accuracy in final answer; units not needed unless change made to metres and/or hours
[5]
# Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differentiate to obtain $kh^n(2+\sqrt{h})^5$ | M1 | Any non-zero constants $k$, $n$; condone presence of $-192$ here |
| Obtain $9h^{-\frac{1}{2}}(2+\sqrt{h})^5$ or unsimplified equiv | A1 | Without $-192$ now |
| Divide 150 by their derivative, algebraic or numerical | \*M1 | Using any recognisable attempt at first derivative |
| Substitute $h=1.4$ and evaluate | M1 | Dep \*M; assume appropriate substitution if calculation goes wrong |
| Obtain $0.06$ or $0.060$ or $0.0603$ | A1 | But not greater accuracy in final answer; units not needed unless change made to metres and/or hours |
| **[5]** | | |
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3 The volume, $V$ cubic metres, of water in a reservoir is given by
$$V = 3 ( 2 + \sqrt { h } ) ^ { 6 } - 192 ,$$
where $h$ metres is the depth of the water. Water is flowing into the reservoir at a constant rate of 150 cubic metres per hour. Find the rate at which the depth of water is increasing at the instant when the depth is 1.4 metres.
\hfill \mbox{\textit{OCR C3 2015 Q3 [5]}}