Mathematics Problems

1. On a trip to the supermarket you buy an economy pack of Soft-4-U toilet tissue. At home you read this information on the side of the pack:

12 rolls 2 ply tissue
Average 225 sheets per roll
Sheet size 125 mm x 110 mm

This sets you thinking about the size of the toilet rolls before you unpack them.

(a) Assuming each sheet has a thickness of 1 mm, you calculate exactly the volume of tissue in any given toilet roll in mm3.

(b) You sketch a diagram of one roll showing its height 110 mm, its outer radius R mm and the radius, 25 mm, of its hollow tabular core. (You have made this measurement from a similar tube.)

(c) Treating the toilet roll as a hollow cylinder and using your answer to (a), you calculate the value of R correct to one decimal place.

When you unpack the toilet rolls you can see at once that the answer you calculated in (c) is too big! You measure it to find that actually R = 60 mm and you realise that you overestimated the thickness of the toilet tissue.

(d) Now treating the toilet roll as a hollow cylinder of height 110 mm, inner radius 25 mm and outer radius 60 mm you set about calculating the thickness, t mm, of each sheet of tissue, correct to 2 decimal places.

2. A car manufacturing plant performs tests on the prototype of a new model engine. The test car runs a distance of 80 km each time at constant speed and the fuel consumption is recorded. The following data is generated:

Speed
S (kmh-1) 40 50 75 90 110
Fuel consumption
F (litres) 13.2 11.6 8.8 9.1 13.5

It is important for the manufacturers to know the most fuel-economical speed of the engine and its minimum fuel consumption over a distance of 80 km at constant speed.

As one of their research and development team, they pass the data to you for analysis because you are the mathematician on that team. These are the most likely steps you would take to solve the problem, increasing the probable level of accuracy each time.

(a) You plot, by hand, a graph of F (vertically) against S (horizontally) using as large a scale as possible for 40 < S <110 and 8 < F < 14 joining the points with as smooth a curve as possible.
[Hint: You need not start your axes from (0, 0)]

(b) You read from the graph, as carefully as possible, the minimum value of F and its corresponding value of S.

(c) You then feed the original data into a suitable software package in order to find the appropriate 'mathematical model' of the problem. The result you are given is:

F = 0.004 S^2 - 0.61 S + 32

Knowing that this could improve your estimate in (b), you tabulate the values of F, correct to 3 decimal places, for 74 < S < 78 in steps of 1 unit.

(d) State the minimum value of F and corresponding value of S as indicated by (c).

(e) You then realise that, from the model given you in (c), you can find the exact value of S by solving the equation

0.008 S - 0.61 = 0

So you do this to find S and then use (c) to find the corresponding value of F correct to 3 significant figures.

3. A ship is carrying 50 tonnes of oil. At the refinery terminal the oil can be pumped out at a rate of m tonnes per minute.

(a) Write down an expression for the time, in minutes, taken to empty the ship.

If the oil is heated up an extra 0.5 tonnes can be emptied out per minute.

(b) Write down an expression for the time taken to empty the ship when the oil is hot.

The difference between these times is 4 minutes.

(c) Write down an equation in m that expresses this using your answers to parts (a) and (b).

(d) Using algebra, show that the equation in (c) becomes

4 m^2 + 2 m - 25 = 0

(e) Solve this equation to find the value of m.

(f) Hence, find the time to empty this ship when the oil is cold.