91Ó°ÊÓ

Peter Minuit of the Dutch West India Company purchased Manhattan Island from the natives living there in 1626 for \(\$ 24\) worth of merchandise. Assuming an exponential rate of inflation of \(5 \%,\) how much will Manhattan be worth in \(2020 ?\)

The population of the United States in 1776 was about 2,508,000 In the country's bicentennial year, the population was about 216,000,000 a) Assuming an exponential model, what was the growth rate of the United States through its bicentennial year? b) Is exponential growth a reasonable assumption? Explain.

The intensity of an earthquake is given by \(I=I_{0} 10^{R},\) where \(R\) is the magnitude on the Richter scale and \(I_{0}\) is the minimum intensity, at which \(R=0,\) used for comparison. a) Find \(I,\) in terms of \(I_{0},\) for an earthquake of magnitude 7 on the Richter scale. b) Find \(I\), in terms of \(I_{0},\) for an earthquake of magnitude 8 on the Richter scale. c) Compare your answers to parts (a) and (b). d) Find the rate of change dI/dR. e) Interpret the meaning of \(d I / d R\)

The number of women graduating from 4 -yr colleges in the United States grew from \(1930,\) when 48,869 women earned a bachelor's degree, to 2005, when approximately 832,000 women received such a degree. Find an exponential function that fits the data, and the exponential growth rate, rounded to the nearest hundredth of a percent.

Solve for \(t\). $$e^{-t}=0.01$$

The Hullian learning model asserts that the probability p of mastering a task after \(t\) learning trials is approximated by \(p(t)=1-e^{-k t}\) where \(k\) is a constant that depends on the task to be learned. Suppose that a new dance is taught to an aerobics class. For this particular dance, the constant \(k=0.28\) a) What is the probability of mastering the dance's steps in 1 trial? 2 trials? 5 trials? 11 trials? 16 trials? 20 trials? b) Find the rate of change, \(p^{\prime}(t)\) c) Sketch a graph of the function.

The Beer-Lambert Law. A beam of light enters a medium such as water or smoky air with initial intensity \(I_{0} .\) Its intensity is decreased depending on the thickness (or concentration) of the medium. The intensity 1 at a depth (or concentration) of \(x\) units is given by $$I=I_{0} e^{-\mu x}$$ The constant \(\mu\left(" m u^{n}\right),\) called the coefficient of absorption, varies with the medium. Use this law for Exercises 52 and \(53 .\) Light through sea water. Sea water has \(\mu=1.4\) and \(x\) is measured in meters. What would increase cloudiness more - dropping \(x\) from 2 m to 5 m or dropping \(x\) from \(7 \mathrm{m}\) to \(10 \mathrm{m} ?\) Explain.

A quantity \(Q_{1}\) grows exponentially with a doubling time of 1 yr. A quantity \(Q_{2}\) grows exponentially with a doubling time of 2 yr. If the initial amounts of \(Q_{1}\) and \(Q_{2}\) are the same, how long will it take for \(Q_{1}\) to be twice the size of \(Q_{2} ?\)

Explain how the Rule of 70 could be useful to someone studying inflation.

Differentiate. $$g(x)=(\ln x)^{4}$$ (Hint: Use the Extended Power Rule.)