INTO Manchester 16/17
8th May – 22nd May
Must be HANDED IN by 12 noon on the 22nd May.
Home assignment
The marks for each part are shown on the paper.

IMaths Part 2 (Engineering) – IFYME002

Maths Part 2 –Engineering
The Task
When carrying out this task, take great care to check that figures are entered accurately into your calculator or computer. Just one mistake could make a big difference to your results and cause you to lose many marks.
Marks for each part are shown in square brackets.
This task involves applying calculus to a component which is used for engineering purposes. You will then go on to use the Normal distribution to find acceptable sizes of this component and to see if a machine which makes pins is likely to be working properly
Figure 1 shows part of the curve y=1/√(1-x^2 ) between x=0 and ??=3/4.

Figure- 1

The area bound by the curve, the ??− and ??− axes and the line x=3/4 (which is shaded on the diagram) represents a thin sheet of metal.
Divide this area into two strips and use the trapezium rule to find an estimate of its area. [ 3 ]
Repeat part [a] with the area divided into four strips. [ 3 ]
Find a more accurate area by using Excel/Autograph with increasing numbers of strips (at least 100). What do you notice? [ 5 ]

a) Show that:1-〖sin〗^2 θ=〖cos〗^2 θ. [ 2 ]
b) By using the substitutionx=sinθ, show that ∫▒〖1/(√(1-x^2 ) ) dx=〖sin〗^(-1) x+c〗 [ 4 ]
c) Hence evaluate ∫_0^(3/4)▒〖1/(√(1-x^2 ) ) dx〗 [ 2 ]
d) Compare this answer to your answer to part [c] in question 1. [ 1 ]

Figure-2 shows the area bound by the curvey=sinx , the ??− axis, and the linex=π/2 and which represents a thin sheet of metal.

The area bound by the curve, the ??− axis and the line x=π/2is rotated about the ??− axis. Find the exact volume of the solid formed. [ 4]

This solid has a rod fixed through its centre and is used in a machine. The solid has to pass through a cylinder and its measurements must be accurate.
The diameter of its circular end has a mean of 90 mm and standard deviation 3 mm. You can assume that the diameters of these solids follow a Normal distribution.
If the diameter is more than 94 mm, the solid is rejected as being too large. If the diameter is less than 87 mm, it is rejected as being too small.

What percentage of the solids will be accepted? Give your answer to 2 significant figures.
(In this question, one mark will be awarded for the correct use of significant figures.) [ 8]

When the machine is in operation, the air temperature inside the cylinder rises. After 1 hour the temperature reaches its maximum and for safety, the machine is turned off and the temperature cools down. The rate of change of temperature, ??°C, of the cooling machine is represented by the differential equation

Where ?? is the time in minutes from when the machine is switched off and ?? is a constant.

Solve the differential equation, giving your answer in the form T=Ae^(-kt) where ?? is a constant. [ 3 ]
Initially, the temperature was 80°C. State the value of ??. [ 1 ]
After 20 minutes, the temperature was 40°C. Show that the value of ?? is 0.035 to two significant figures. [ 2 ]
Find the temperature after 30 minutes. [ 2 ]

Coca-Cola Enterprises in Milton Keynes site has the world’s first 2,000-can-per-minute filling machine for sparkling soft drinks and has recently installed a new canning line to increase production levels, especially of the 80ml and 500ml cans of the company’s new Relentless energy drink.

For the 500ml cans of the new drink, the standard deviation is known to be 5 ml. The volumes of the liquid can be assumed to follow a Normal distribution. There is a need to check if the new canning line is operating correctly. (Kable (2015) Coca-Cola Canning Line, United Kingdom [online] available at: : [Accessed 31st Oct 2016])

A random sample of 30 cans is selected. The mean volume was found to be 497 ml. Set up a 95% confidence interval of the mean volume of the cans filled by the machine. [ 3 ]

Should the machine be stopped and adjusted? Give a reason. [ 1 ]

Summary and Conclusion
In less than 100 words, write a summary on what improvements you would make in checking the machine in question 6. In particular, state how you would do things differently to try to obtain more reliable results. [ 6 ]