MATH 136 – Calculus 2 Group Discussion Day 5 – Applications of differential Equations

I. A rare mineral deposit was formed in the collision of a meteorite with the earth some time long in the past. The deposit originally contained some amount of the Uranium isotope U^{238}, which is radioactive with half-life 4.51 × 10^{9} years, but none of the lead isotope P b^{207}, which is the end decay product of U^{238}. The ratio of the number of U^{238} atoms to the number of P b^{207} in the mineral deposit today is 0.9. When did the collision with the meteorite occur?

II. Just before midday, the body of an apparent homicide victim is found in a room that is kept at a constant temperature of 70 degrees F. At 12 noon the temperature of the body is measured to be 80 degrees F and at 1 pm, the temperature has decreased to 75 degrees F. Assume that the temperature of the body at the time of death was 98.6 degrees F and that the body cooled according to Newton’s law. When did the murder occur?

III. Suppose that a water tank has a hole at the bottom and water is draining out. If y(t) denotes the depth of the water at time t, then Torricelli’s Law implies that y(t) will satisfy the differential equation

A(y)\frac{dy}{dt} = −a\sqrt{2gy}

where

• g is the acceleration of gravity near the surface of the earth (≐ 32 in feet per seconds squared),

• A is the cross-section area of the tank at depth y, and

• a is a constant (it represents the area of the hole).

A) If the tank has the shape of the solid obtained by rotating some curve x = g(y) about the y-axis, determine g(y) so that the depth of the water in the tank will fall at a constant rate.

B) For what would such a tank be useful? (Hint: Look up the term clepsydra on Wikipedia. An “outflow” clepsydra formed from a tank of this shape would have a special property.)