91Ó°ÊÓ

Cesium- 137 has a decay rate of 2.3\% per year. Suppose cesium-137 is released into the atmosphere for \(20 \mathrm{yr}\) at a rate of \(1 \mathrm{lb}\) per year. How much cesium- 137 will be present in the atmosphere after 20 yr?

(a) write a differential equation that models the situation, and (b) find the general solution. If an initial condition is given, find the particular solution. Recall that when \(y\) is directly proportional to \(x,\) we have \(y=k x\), and when \(y\) is inversely proportional to \(x,\) we have \(y=k / x,\) where \(k\) is the constant of proportionality. In these exercises, let \(k=1\). The rate of change of \(y\) with respect to \(x\) is inversely proportional to the square root of \(y\).

The capitalized cost, \(c,\) of an asset over its lifetime is the total of the initial cost and the present value of all maintenance expenses that will occur in the future. It is computed with the formula $$ c=c_{0}+\int_{0}^{L} m(t) e^{-k t} d t $$ where \(c_{0}\) is the initial cost of the asset, \(L\) is the lifetime (in years), \(k\) is the interest rate (compounded continuously), and \(m(t)\) is the annual cost of maintenance. Find the capitalized cost under each set of assumptions. $$ \begin{array}{l} c_{0}=\$ 300,000, k=5 \%, m(t)=\$ 30,000+\$ 500 t, \\ L=20 \end{array} $$

Hanna's Hat Company's marginal profit, \(P\), as a function of its total cost, \(C,\) is given by \(\frac{d P}{d C}=\frac{-200}{(C+3)^{3 / 2}}\) a) Find the profit function, \(P(C),\) if \(P=\$ 10\) when \(C=\$ 61\) b) At what cost will the firm break even \((P=0)\) ?

In a normal distribution with \(\mu=-15\) and \(\sigma=0.4\) find the \(x\) -value that corresponds to the a) 46 th percentile b) 92 nd percentile

Test score distribution. In a large class, student test scores had a mean of \(\mu=76\) and a standard deviation \(\sigma=7\) a) The top \(12 \%\) of students got an \(A\). Find the minimum score needed to get an \(\mathrm{A}\) (round to the appropriate integer). b) The top \(75 \%\) of students passed. Find the minimum score needed to pass (round to the appropriate integer).