Determine how long it will take the milk to completely

Question

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=1}Σⁿ 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.

QUESTION 3:

The State Government requires a new single train track to be constructed. There is an existing single straight train track, 10√2 kilometres long, running in a north easterly direction from Riemann's Pier. The new track has to continue smoothly from this existing track and continue until Newton's Jetty which is 30 kilometres due east of Riemann's Pier.

Newton's Jetty and Riemann's Pier lie on the edge of the northern bank of Euler's River which runs a straight course.

Due to budgetary constraints the Government wishes the new piece of track to be as short as possible. However, the Department of Experimental Sciences holds the land between the river and the track (including the existing and proposed sections). They require exactly 250 square kilometres in which to maintain their experimental programmes.

(a) On the xy plane model the existing track with the function f(x) = x which starts at the origin (Riemann's Pier) and continues to the point (10,10). Determine g(x), giving reasons) which models the proposed track starting at (10,10) and continues to (30,0) representing Newton's Jetty. The northern bank of Euler's River is, of course, represented by the x-axis.

Note that g is required to be a continuous function that satisfies all of the following properties:

(α) g(10) = 10, g(30) = 0 ,

(β) g is non-negative, (to avoid the track plunging into the river of course)

(γ) The area between the graph of g and the x-axis from x = 10 to x = 30 is equal to 200:

₁₀∫³⁰ g(x) dx = 200,

(δ) g has to connect "smoothly" to f at (10,10) and continue "smoothly" to (30,0).

(b) Use Wolfram Alpha to obtain a graph of the function you found in part (a).

(c) What, exactly, were we assuming when we modelled the track on the xy plane in part (a)?

(d) Calculate the length of the proposed section of train track. The arc-length for your function g must be found using

L = ₁₀∫³⁰√(1 + g'(x)]²dx).

you must calculate the resulting integral as far as you possibly can by using the exact integration techniques presented in 300673 Mathematics 1B. If necessary, to obtain a decimal approximation, you will be required to use Simpson's Rule as defined in Lecture 7. As a guide only you can check your calculation using the arc-length function in "Wolfram Alpha".

Each member of the group that finds the proposed track with the smallest length satisfying properties (α), (β), (γ) and (δ), with a resonable interpretation of "smooth" in (δ), will be awarded bonus marks (up to a maximum of 20% of the raw mark).