1.02r Proportional relationships: and their graphs

3 It is given that \(x\) is proportional to the product of the square of \(y\) and the positive square root of \(z\). When \(y = 2\) and \(z = 9 , x = 30\).

1. Write an equation for \(x\) in terms of \(y\) and \(z\).
2. Find the value of \(x\) when \(y = 3\) and \(z = 25\).

6 The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(v \mathrm {~ms} ^ { - 1 }\). When \(v = 3.6 , P = 50\).
Determine the wind speed that will give a power output of 225 watts.

5 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-04_700_727_260_242} The diagram shows a curve \(C\) for which \(y\) is inversely proportional to \(x\). The curve passes through the point \(\left( 1 , - \frac { 1 } { 2 } \right)\).

1. 1. Determine the equation of the gradient function for the curve \(C\).
   2. Sketch this gradient function on the axes in the Printed Answer Booklet.
2. The diagram indicates that the curve \(C\) has no stationary points. State what feature of your sketch in part (a)(ii) corresponds to this.
3. The curve \(C\) is translated by the vector \(\binom { - 2 } { 0 }\). Find the equation of the curve after it has been translated.

11 The intensity of the sun's radiation, \(y\) watts per square metre, and the average distance from the sun, \(x\) astronomical units, are shown in Fig. 11 for the planets Mercury and Jupiter. \begin{table}[h] \end{table} The intensity \(y\) is proportional to a power of the distance \(x\).

1. Write down an equation for \(y\) in terms of \(x\) and two constants.
2. Show that the equation can be written in the form \(\ln y = a + b \ln x\).
3. In the Printed Answer Booklet, complete the table for \(\ln x\) and \(\ln y\) correct to 4 significant figures.
4. Use the values from part (iii) to find \(a\) and \(b\).
5. Hence rewrite your equation from part (i) for \(y\) in terms of \(x\), using suitable numerical values for the constants.
6. Sketch a graph of the equation found in part (v).
7. Earth is 1 astronomical unit from the sun. Find the intensity of the sun's radiation for Earth.

11 David puts a block of ice into a cool-box. He wishes to model the mass \(m \mathrm {~kg}\) of the remaining block of ice at time \(t\) hours later. He finds that when \(t = 5 , m = 2.1\), and when \(t = 50 , m = 0.21\).

1. David at first guesses that the mass may be inversely proportional to time. Show that this model fits his measurements.
2. Explain why this model
   1. is not suitable for small values of \(t\),
   2. cannot be used to find the time for the block to melt completely. David instead proposes a linear model \(m = a t + b\), where \(a\) and \(b\) are constants.
3. Find the values of the constants for which the model fits the mass of the block when \(t = 5\) and \(t = 50\).
4. Interpret these values of \(a\) and \(b\).
5. Find the time according to this model for the block of ice to melt completely.

11 In this question you must show detailed reasoning.
The variables \(x\) and \(y\) are such that \(\frac { \mathrm { dy } } { \mathrm { dx } }\) is directly proportional to the square root of \(x\).
When \(x = 4 , \frac { d y } { d x } = 3\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) in terms of \(x\). When \(\mathrm { x } = 4 , \mathrm { y } = 10\).
2. Find \(y\) in terms of \(x\).

14

1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
2. Prove that \(h = \sqrt { a b }\), as given in line 24 .

1 A curve for which \(y\) is inversely proportional to \(x\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-4_824_1125_561_242} Find the equation of the curve.

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.

A craft artist is producing items and selling them in a local market. The selling price, £P, of an item is inversely proportional to the number of items produced, \(n\)

1. When \(n = 10\), \(P = 24\) Show that \(P = \frac{240}{n}\) [1 mark]
2. The production cost, £C, of an item is inversely proportional to the square of the number of items produced, \(n\) When \(n = 10\), \(C = 60\) Find the set of values of \(n\) for which \(P > C\) [4 marks]
3. Explain the significance to the craft artist of the range of values found in part (b). [1 mark]

A is inversely proportional to B. B is inversely proportional to the square of C. When A is 2, C is 8. Find C when A is 12. [3]

The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(v\)ms\(^{-1}\). When \(v = 3.6\), \(P = 50\). Determine the wind speed that will give a power output of 225 watts. [3]