## Exercises

1. Suppose element Z has a half-life of 2 years.

(a) Choose an initial amount for the element. Find the amount as a function of time, and plot this function.

(b) How long is it until the amount is of what it was initially?

(c) How long is it until the amount is of what it was initially? Is there a quick way to answer this question?

2. Decibels:

(a) If a sound source is r times farther from point A than point B, the intensity at point A is times what it is at point B. (For example if you move ten times closer to a sound source, the intensity you encounter is 100 times greater.) Find and plot the difference between the decibel measurements at points A and B, as a function of the ratio of distances r.

(b) If 60 decibels are measured at point A, and 80 decibels are measured at point B, Find the ratio of the intensities at points A and B.

3. Consider the logistic growth differential equation

Separating the variables yields the equation

(a) Use maple to evaluate these integrals. Then use maple's results to solve the differential equation for y as a function of x. (Remember that maple does not include a constant of integration in its answer, so you will have to include it yourself. Note, also, that maple does not include absolute-value signs when integrating , so you may have to modify the arguments of logarithms, to make sure that you are taking the log of a positive number. Be sure to explain clearly what you are doing.)

(b) Explain how your answers to (a) may be written in the form

where A is some positive constant.

(c) Suppose at time t=0 the population of bacteria in a culture is 1000, and the limited environment can support no more than 100,000 bacteria. After 1 hour the population is 5,000. Find the values of A and k. Then plot y as a function of t. What is the population when the rate of growth is most rapid?