QUESTION 1:

(a) (i) Show that

_i=1Σⁿ e^i/n = (e^1/n(1-e))/(1-e^1/n).

Hint: The series _i=1Σⁿ e^i/n is a geometric series. Use your knowledge of geometric series from high school to evaluate this series.

(ii) Use l'Hopital's rule to evaluate

lim_t→∞ [t(1 - e^1/t)].

Hint: Recall the methods using l'Hopital's rule from Mathematics 1A where we had limits in the form "∞ x 0".

(iii) Recall the definition of the definite integral:

_a∫^b f (x) dx = lim_n→∞ i=1Σⁿ f(x_i*)Δx

where the notation is that which is used in Lecture 1. Use the definition to find the area between f (x) = e^x and the x-axis between x = 0 and x = 1. You may require your results from parts (i) and (ii) above.

(b) The manager of the local "McMuckies" fast food outlet has found that the average waiting time that her customers have to wait for service is 3 minutes.

(i) Find the percentage of customers who have to wait more than 4 minutes.

(ii) Find the percentage of customers that are served within the first 2 minutes.

(iii) Sales have been falling, so the manager wants to advertise that anybody who isn't served within a certain number of minutes gets a free burger. But she doesn't want to give away free burgers to more than 3% of her customers. What should the advertisement say?

QUESTION 2:

Recall Torricelli's law for a liquid draining from a tank:

A(h) dh/dt = -k√h

where the notation is that used in Lecture 6.

(a) Show that, in general, the differential equation in Torricelli's law may be solved by separation of variables. State any assumptions you have used.

(b) (i) A dairy farmer has a large vat three-quarters filled with milk. The vat is in the shape of a trapezoidal prism with length 10 metres, perpendicular height 4 metres, lower width 4 metres and upper width 5 metres. See the diagram. The drain at the bottom of the tank is opened. Use Torricelli's law with k = 0.03 to determine the depth of the milk at any instant after the drain has been opened.

(ii) Use Wolfram Alpha to obtain a plot of h versus t.

(iii) Determine how long it will take the milk to completely drain from the vat.

(c) Is it possible to design a tank such that the depth of the liquid, h, will be a linear function of time, t ?

Provide assumptions and detailed calculations either proving the impossibility of such a tank or showing the dimensions of a tank for which this is possible.

Note that a group which is able to provide sound mathematical reasoning will gain more marks than a group using pure guesswork.