Chain rule with three variables

5 \includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-06_462_878_258_635} The dimensions of a cuboid are \(x \mathrm {~cm} , 2 x \mathrm {~cm}\) and \(4 x \mathrm {~cm}\), as shown in the diagram.

1. Show that the surface area \(S \mathrm {~cm} ^ { 2 }\) and the volume \(V \mathrm {~cm} ^ { 3 }\) are connected by the relation $$S = 7 V ^ { \frac { 2 } { 3 } }$$
2. When the volume of the cuboid is \(1000 \mathrm {~cm} ^ { 3 }\) the surface area is increasing at \(2 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the volume at this instant.

7. The volume of a spherical balloon of radius \(r \mathrm {~cm}\) is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\). The volume of the balloon increases with time \(t\) seconds according to the formula $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1000 } { ( 2 t + 1 ) ^ { 2 } } , \quad t \geqslant 0$$
2. Using the chain rule, or otherwise, find an expression in terms of \(r\) and \(t\) for \(\frac { \mathrm { d } r } { \mathrm {~d} t }\).
3. Given that \(V = 0\) when \(t = 0\), solve the differential equation \(\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1000 } { ( 2 t + 1 ) ^ { 2 } }\), to obtain \(V\) in terms of \(t\).
4. Hence, at time \(t = 5\),
   1. find the radius of the balloon, giving your answer to 3 significant figures,
   2. show that the rate of increase of the radius of the balloon is approximately \(2.90 \times 10 ^ { - 2 } \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).

3 The driving force \(F\) newtons and velocity \(v \mathrm {~km} \mathrm {~s} ^ { - 1 }\) of a car at time \(t\) seconds are related by the equation \(F = \frac { 25 } { v }\).

1. Find \(\frac { \mathrm { d } F } { \mathrm {~d} v }\).
2. Find \(\frac { \mathrm { d } F } { \mathrm {~d} t }\) when \(v = 50\) and \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 1.5\).

9 Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, \(B\), and the number of wildflower plants, \(F\), at time \(t\) weeks after the start of the study. In the model \(B = 20 + 2 t + \cos 3 t\) and \(F = 50 \mathrm { e } ^ { 0.1 t }\). The model assumes that \(B\) and \(F\) can be treated as continuous variables.

1. State the meaning of \(\frac { \mathrm { d } B } { \mathrm {~d} F }\).
2. Determine \(\frac { \mathrm { d } B } { \mathrm {~d} F }\) when \(t = 4\).
3. Suggest a reason why this model may not be valid for values of \(t\) greater than 12 .

12 Boyle's Law states that when a gas is kept at a constant temperature, its pressure \(P\) pascals is inversely proportional to its volume \(V \mathrm {~m} ^ { 3 }\). When the volume of a certain gas is \(80 \mathrm {~m} ^ { 3 }\), its pressure is 5 pascals and the rate at which the volume is increasing is \(10 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the pressure is decreasing at this volume.

\includegraphics{figure_2} The diagram shows the curve \(y = 2x^2\) and the points \(X(-2, 0)\) and \(P(p, 0)\). The point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.

1. Express the area, \(A\), of triangle \(XPQ\) in terms of \(p\). [2]

The point \(P\) moves along the \(x\)-axis at a constant rate of 0.02 units per second and \(Q\) moves along the curve so that \(PQ\) remains parallel to the \(y\)-axis.

1. Find the rate at which \(A\) is increasing when \(p = 2\). [3]

The driving force \(F\) newtons and velocity \(v\) km s\(^{-1}\) of a car at time \(t\) seconds are related by the equation \(F = \frac{25}{v}\).

1. Find \(\frac{dF}{dv}\). [2]
2. Find \(\frac{dF}{dt}\) when \(v = 50\) and \(\frac{dv}{dt} = 1.5\). [3]

The curve \(y = x^3\) intersects the line \(y = kx\), \(k > 0\), at the origin and the point \(P\). The region bounded by the curve and the line, between the origin and \(P\), is denoted by \(R\).

1. Show that the area of the region \(R\) is \(\frac{1}{6}k^3\). [3]

The line \(x = a\) cuts the region \(R\) into two parts of equal area.

1. Show that \(k^3 - 6a^2k + 4a^3 = 0\). [3]

The gradient of the line \(y = kx\) increases at a constant rate with respect to time \(t\). Given that \(\frac{dk}{dt} = 2\),

1. determine the value of \(\frac{da}{dt}\) when \(a = 1\) and \(k = 2\), [4]
2. determine the value of \(\frac{da}{dt}\) when \(a = 1\) and \(k = 2\), expressing your answer in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are integers. [5]