91Ó°ÊÓ

A certain moon rock was found to contain equal numbers of potassium and argon atoms. Assume that all the argon is the result of radioactive decay of potassium (its half-life is about \(1.28 \times 10^{9}\) years) and that one of every nine potassium atom disintegrations yields an argon atom. What is the age of the rock, measured from the time it contained only potassium?

When sugar is dissolved in water, the amount \(A\) that remains undissolved after \(t\) minutes satisfies the differential equation \(d A / d t=-k A(k>0) .\) If \(25 \%\) of the sugar dissolves after 1 min, how long does it take for half of the sugar to dissolve?

Problems deal with a shallow reservoir that has a one-square-kilometer water surface and an average water depth of 2 meters. Initially it is filled with fresh water, but at time \(t=0\) water contaminated with a liquid pollutant begins flowing into the reservoir at the rate of 200 thousand cubic meters per month. The well-mixed water in the reservoir flows out at the same rate. Your first task is to find the amount \(x(t)\) of pollutant (in millions of liters) in the reservoir after 1 months. The incoming water has a pollutant concentration of \(c(t)=10\) liters per cubic meter \(\left(\mathrm{L} / \mathrm{m}^{3}\right)\). Verify that the graph of \(x(t)\) resembles the steadily rising curve in Fig. 1.5.9, which approaches asymptotically the graph of the equilibrium solution \(x(t)=20\) that corresponds to the reservoir's long-term pollutant content. How long does it take the pollutant concentration in the reservoir to reach \(10 \mathrm{~L} / \mathrm{m}^{3} ?\)

A certain piece of dubious information about phenylethylamine in the drinking water began to spread one day in a city with a population of 100,000 . Within a week, 10,000 people had heard this rumor. Assume that the rate of increase of the number who have heard the rumor is proportional to the number who have not yet heard it. How long will it be until half the population of the city has heard the rumor?

An accident at a nuclear power plant has left the surrounding area polluted with radioactive material that decays naturally. The initial amount of radioactive material present is 15 su (safe units), and 5 months later it is still 10 su. (a) Write a formula giving the amount \(A(t)\) of radioactive material (in su) remaining after \(t\) months. (b) What amount of radioactive material will remain after 8 months? (c) How long-total number of months or fraction thereof-will it be until \(A=1 \mathrm{su}\), so it is safe for people to return to the area?

Suppose a uniform flexible cable is suspended between two points \((\pm L, H)\) at equal heights located symmetrically on either side of the \(x\) -axis (Fig. 1.4.12). Principles of physics can be used to show that the shape \(y=y(x)\) of the hanging cable satisfies the differential equation $$a \frac{d^{2} y}{d x^{2}}=\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}$$ where the constant \(a=T / \rho\) is the ratio of the cable's tension \(T\) at its lowest point \(x=0\) (where \(y^{\prime}(0)=0\) ) and its (constant) linear density \(\rho\). If we substitute \(v=d y / d x\), \(d v / d x=d^{2} y / d x^{2}\) in this second-order differential equation, we get the first-order equation $$ a \frac{d v}{d x}=\sqrt{1+v^{2}} $$ Solve this differential equation for \(y^{\prime}(x)=v(x)=\) \(\sinh (x / a)\). Then integrate to get the shape function $$y(x)=a \cosh \left(\frac{x}{a}\right)+C$$ of the hanging cable. This curve is called a catenary, from the Latin word for chain.