Related rates with explicitly given non-geometric algebraic relationships

9 The volume \(V \mathrm {~m} ^ { 3 }\) of a large circular mound of iron ore of radius \(r \mathrm {~m}\) is modelled by the equation \(V = \frac { 3 } { 2 } \left( r - \frac { 1 } { 2 } \right) ^ { 3 } - 1\) for \(r \geqslant 2\). Iron ore is added to the mound at a constant rate of \(1.5 \mathrm {~m} ^ { 3 }\) per second.
[0pt]

1. Find the rate at which the radius of the mound is increasing at the instant when the radius is 5.5 m . [3]
2. Find the volume of the mound at the instant when the radius is increasing at 0.1 m per second.

10. \begin{figure}[h] \end{figure} Figure 4 shows a right circular cylindrical rod which is expanding as it is heated.
At time \(t\) seconds the radius of the rod is \(x \mathrm {~cm}\) and the length of the rod is \(6 x \mathrm {~cm}\).
Given that the cross-sectional area of the rod is increasing at a constant rate of \(\frac { \pi } { 20 } \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\), find the rate of increase of the volume of the rod when \(x = 2\) Write your answer in the form \(k \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) where \(k\) is a rational number.

4. \begin{figure}[h] \end{figure} A regular icosahedron of side length \(x \mathrm {~cm}\), shown in Figure 1, is expanding uniformly. The icosahedron consists of 20 congruent equilateral triangular faces of side length \(x \mathrm {~cm}\).

1. Show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the icosahedron is given by $$A = 5 \sqrt { 3 } x ^ { 2 }$$ Given that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the icosahedron is given by $$V = \frac { 5 } { 12 } ( 3 + \sqrt { 5 } ) x ^ { 3 }$$
2. show that \(\frac { \mathrm { d } V } { \mathrm {~d} A } = \frac { ( 3 + \sqrt { 5 } ) x } { 8 \sqrt { 3 } }\) The surface area of the icosahedron is increasing at a constant rate of \(0.025 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\)
3. Find the rate of change of the volume of the icosahedron when \(x = 2\), giving your answer to 2 significant figures.

4. \begin{figure}[h] \end{figure} A cone, shown in Figure 2, has

• fixed height 5 cm
• base radius \(r \mathrm {~cm}\)
• slant height \(l \mathrm {~cm}\)
  1. Find an expression for \(l\) in terms of \(r\)

Given that the base radius is increasing at a constant rate of 3 cm per minute,

find the rate at which the total surface area of the cone is changing when the radius of the cone is 1.5 cm . Give your answer in \(\mathrm { cm } ^ { 2 }\) per minute to one decimal place.
[0pt] [The total surface area, \(S\), of a cone is given by the formula \(S = \pi r ^ { 2 } + \pi r l\) ]

3. \begin{figure}[h] \end{figure} Figure 2 shows a right circular cylindrical metal rod which is expanding as it is heated. After \(t\) seconds the radius of the rod is \(x \mathrm {~cm}\) and the length of the rod is \(5 x \mathrm {~cm}\). The cross-sectional area of the rod is increasing at the constant rate of \(0.032 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\).

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when the radius of the rod is 2 cm , giving your answer to 3 significant figures.
2. Find the rate of increase of the volume of the rod when \(x = 2\).
   \section*{LU}

4 When the gas in a balloon is kept at a constant temperature, the pressure \(P\) in atmospheres and the volume \(V \mathrm {~m} ^ { 3 }\) are related by the equation $$P = \frac { k } { V }$$ where \(k\) is a constant. [This is known as Boyle's Law.]
When the volume is \(100 \mathrm {~m} ^ { 3 }\), the pressure is 5 atmospheres, and the volume is increasing at a rate of \(10 \mathrm {~m} ^ { 3 }\) per second.

1. Show that \(k = 500\).
2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} V }\) in terms of \(V\).
3. Find the rate at which the pressure is decreasing when \(V = 100\).

4 A piston can slide inside a tube which is closed at one end and encloses a quantity of gas (see Fig. 4). \begin{figure}[h] \end{figure} The pressure of the gas in atmospheric units is given by \(p = \frac { 100 } { x }\), where \(x \mathrm {~cm}\) is the distance of the piston from the closed end. At a certain moment, \(x = 50\), and the piston is being pulled away from the closed end at 10 cm per minute. At what rate is the pressure changing at that time?

6 When the gas in a balloon is kept at a constant temperature, the pressure \(P\) in atmospheres and the volume \(V \mathrm {~m} ^ { 3 }\) are related by the equation $$P = \frac { k } { V } ,$$ where \(k\) is a constant. [This is known as Boyle's Law.]
When the volume is \(100 \mathrm {~m} ^ { 3 }\), the pressure is 5 atmospheres, and the volume is increasing at a rate of \(10 \mathrm {~m} ^ { 3 }\) per second.

1. Show that \(k = 500\).
2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} V }\) in terms of \(V\).
3. Find the rate at which the pressure is decreasing when \(V = 100\).

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.

A particle \(P\) of mass \(2 \text{ kg}\) moving on a horizontal straight line has displacement \(x \text{ m}\) from a fixed point \(O\) on the line and velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\). The only horizontal force acting on \(P\) is a variable force \(F \text{ N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).

1. Find an expression for \(x\) in terms of \(t\). [4]