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Problems in Calculus, Combinatorics and Functional Analysis

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Problem 1:

A tank contains 100 gal of brine may by dissolving 80 lb of salt in water. Pure water runs into the tank at the rate of 4gal/min and the mixture, which is kept uniform by stirring, runs out at the same rate. Find the amount y(t) of salt in the tank at any time t.

Problem 2:
For a continuous and onto function F from [0,2] to [0,4]. Show that F(x) = 2x for some .

Problem 3:
2011 students standing in a row, each of a different height. In how many ways can you arrange them such that there are never three of them (not necessarily next to each other) with the tallest in the middle? Explain.

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Solution Summary

We solve a word problem in calculus, a problem in functional analysis, and a problem in combinatorics.

Solution Preview

1. There are always 100 gallons of brine in the tank, so the concentration of brine in the tank at time t (in pounds of salt per gallon of brine) is given by y(t)/100. Now the solution flows out of the tank at a rate of 4 gallons per minute, so the amount of salt per minute flowing out of the tank is y(t)/25. Thus we have

dy/dt = -y/25.

Multiplying both sides by dt/y we obtain

dy/y = -dt/25.

Integrating both sides we obtain

ln y(t) = -t/25 + c.

Exponentiating both sides we obtain

y(t) = C ...

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