In $1980$ the population of alligators on the Kennedy Space Center grounds was estimated to be $1500$. In $2006$ the population had grown to an estimated $6000$. Using the Malthusian law for population growth, estimate the alligator population on the Kennedy Space Center grounds in the year $2020$.

Describe at least one way in which both of the statements in the following pairs could be true.

A: The birth rate (births per $100,000$ people in the population) in a country is increasing.

B: The population of the same country is decreasing.

In $2015$, the U.S. population was about $321$ million. The overall birth rate was $12.5$ births per $1000$ people, and the overall death rate was $8.4$ deaths per $1000$ people. a. Approximately how many births were there in the United States in $2015$? b. About how many deaths were there in the United States in that year? c. Based on births and deaths alone (i.e., not counting immigration), about how much did the U.S. population increase during $2015$? d. Suppose that during $2015$ the U.S. population actually increased by $3.0$ million. Based on this fact and your results from part (c), estimate how many people immigrated to the United States. What proportion of the overall population growth was due to immigration?

A cold beer initially at $35^{\circ}$F warms up to $40^{\circ}$F in $3$ min while sitting in a room of temperature $70^{\circ}$F. How warm will the beer be if left out for $20$ min?

A brine solution of salt flows at a constant rate of 8 L/min into a large tank that initially held 100 L of brine solution in which was dissolved 0.5 kg of salt. The solution inside the tank is kept well stirred and flows out of the tank at the same rate. If the concentration of salt in the brine entering the tank is 0.05 kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.02 kg/L?

Sketch (or computer plot) each of the following functions on the interval (−1, 1) and expand it in a complex exponential series and in a sine-cosine series. (a) $f(x) = x, \ −1 < x < 1.$ (b)

$$f(x)=\left\{\begin{array}{lr} 1+2 x, & -1<x<0 \\ 1-2 x, & 0<x<1 \end{array}\right.$$

(c)

$$f(x)=\left\{\begin{array}{lr} x+x^{2}, & -1<x<0 \\ x-x^{2}, & 0<x<1 \end{array}\right.$$

By experimenting with the improved Euler's method subroutine, find the maximum value over the interval $[ 0, 2 ]$ of the solution to the initial value problem

$$y'-\sin (x_y), \qquad y(0)=2.$$

Where does this maximum value occur? Give answers to two decimal places.

Assume that the gravitational force is constant with g = $9.81$ m/sec$^{2}$ in the $MKS$ system and $g = 32$ ft/sec$^{2}$ in the U.S. Customary System.
When the velocity v of an object is very large, the magnitude of the force due to air resistance is proportional to $v^{2}$ with the force acting in opposition to the motion of the object. A shell of mass $3$ kg is shot upward from the ground with an initial velocity of $500$ m/sec. If the magnitude of the force due to air resistance is $( 0.1 )v^{2}$, when will the shell reach its maximum height above the ground? What is the maximum height?

A sphere of radius $R$, centered at the origin, carries charge density

$$\rho(r,\theta)=k\dfrac{R}{r^2}(R-2r)\sin\theta$$

where k is a constant, and $r$ , $\theta$ are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.

The weight of an object varies inversely as the square of the object's distance from the center of Earth. The radius of Earth is 3960 miles. a. If an astronaut weighs 120 pounds on the surface of Earth, how much does she weigh 6000 miles above the surface of Earth? b. If a miner weighs 200 pounds on the surface of Earth, how much does he weigh 10 miles below the surface of Earth?

Use the fourth-order Runge-Kutta algorithm to approximate the solution to the initial value problem

$$y' = y \cos x,\qquad y(0) = 1,$$

at $x = \pi$. For a tolerance of $\varepsilon = 0.01,$ use a stopping procedure based on the absolute error.

In Problem $13$, if a larger tank with a heat capacity of $1^{\circ}F$ per thousand Btu and a time constant of $72$ hr is used instead (with all other factors being the same), what will be the temperature in the tank after $12$ hr?

Compare the following two pairs of goods:

Coke and Pepsi

Skis and ski bindings

In which case do you expect the indifference curves to be fairly straight? In which case do you expect the indifference curves to be very bowed?

Show that the geodesics of a spherical surface are great circles, i.e., circles whose centers lie at the center of the sphere.