91Ó°ÊÓ

True or False? If true, prove it. If false, find the true answer. If you invest \(\$ 500\), an annual rate of interest of \(3 \%\) yields more money in the first year than a \(2.5 \%\) continuous rate of interest.

True or False? If true, prove it. If false, find the true answer. If given a half-life of \(t\) years, the constant \(k\) for \(y=e^{k t}\) is calculated by \(k=\ln (1 / 2) / t\)

For the following exercises, use \(y=y_{0} e^{\Lambda t}\). If a culture of bacteria doubles in 3 hours, how many hours does it take to multiply by \(10 ?\)

For the following exercises, use \(y=y_{0} e^{\Lambda t}\). If bacteria increase by a factor of 10 in 10 hours, how many hours does it take to increase by \(100 ?\)

For the following exercises, use \(y=y_{0} e^{\Lambda t}\). How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is 5730 years.

For the following exercises, use \(y=y_{0} e^{\Lambda t}\). If a relic contains \(90 \%\) as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is 5730 years.

For the following exercises, use \(y=y_{0} e^{\Lambda t}\). The population of Cairo grew from 5 million to 10 million in 20 years. Use an exponential model to find when the population was 8 million.

For the following exercises, use \(y=y_{0} e^{\Lambda t}\). The populations of New York and Los Angeles are growing at \(1 \%\) and \(1.4 \%\) a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?

For the following exercises, use \(y=y_{0} e^{\Lambda t}\). Suppose the value of \(\$ 1\) in Japanese yen decreases at \(2 \%\) per year. Starting from \(\$ 1=Â¥ 250,\) when will \(\$ 1=Â¥ 1 ?\)

For the following exercises, use \(y=y_{0} e^{\Lambda t}\). The effect of advertising decays exponentially. If \(40 \%\) of the population remembers a new product after 3 days, how long will \(20 \%\) remember it?