The population of the world was about 6.1 billion in 2000. Birth rates around that time ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let’s assume that the carrying capacity for world population is 20 billion.
(a) Write the logistic differential equation for these data.
(Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.)
(b) Use the logistic model to estimate the world population in the year 2010 and compare with the actual population of 6.9 billion.
(c) Use the logistic model to predict the world population in the years 2100 and 2500.
(a) Set up, but do not evaluate, a double integral for the area of the surface with parametric equations x = au cos v, y = bu sin v, z = u2, 0 < u < 2, 0 < v < 2.
(b) Eliminate the parameters to show that the surface is an elliptic paraboloid and set up another double integral for the surface area.
(c) Use the parametric equations in part (a) with a = 2 and b = 3 to graph the surface.
(d) For the case a = 2, b = 3, use a computer algebra system to find the surface area correct to four decimal places.
A solid occupies a region E with surface S and is immersed in a liquid with constant density p. We set up a coordinate system so that the xy-plane coincides with the surface of the liquid, and positive values of z are measured downward into the liquid. Then the pressure at depth z is p = pgz, where t is the acceleration due to gravity (see Section 8.3). The total buoyant force on the solid due to the pressure is given by the surface integralwhere n is the outer unit normal. Use the result of Exercise 31 to show that F = -Wk, where W is the weight of the liquid displaced by the solid. (That F is directed upward because z is directed downward.) The result is Archimedes’ Principle:
The buoyant force on an object equals the weight of the displaced liquid.
Suppose the three coordinate planes are all mirrored and a light ray given by the vector first strikes the xz-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray. (American space scientists used this principle, together with laser beams and an array of corner mirrors on the moon, to calculate very precisely the distance from the earth to the moon.)
The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N45°W at a speed of 50 km/h. (This means that the direction from which the wind blows is 45° west of the northerly direction.) A pilot is steering a plane in the direction N60°E at an airspeed (speed in still air) of 250 km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane.
A molecule of methane, CH4, is structured with the four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the H—C—H combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about 109.5°. [Take the vertices of the tetrahedron to be the points (1, 0, 0), (0, 1, 0), (0, 0, 1), and (1, 1, 1), as shown in the figure. Then the centroid is (12, 12, 12).]
A solid has the following properties. When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a square. If the rays are parallel to the x-axis, its shadow is an isosceles triangle. (In Exercise 12.1.48 you were asked to describe and sketch an example of such a solid, but there are many such solids.) Assume that the projection onto the xz-plane is a square whose sides have length 1.
(a) What is the volume of the largest such solid?
(b) Is there a smallest volume?
The view of the trefoil knot shown in Figure 8 is accurate, but it doesn’t reveal the whole story. Use the parametric equations
x = (2 + cos 1.5t) cos t
y = (2 + cos 1.5t) sin t
z = sin 1.5t
to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the xy-plane has polar coordinates r = 2 + cos 1.5t and θ = t, so r varies between 1 and 3. Then show that z has maximum and minimum values when the projection is halfway between r = 1 and r = 3.
When you have finished your sketch, use a computer to draw the curve with viewpoint directly above and compare with your sketch. Then use the computer to draw the curve from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around the curve. (Use the tubeplot command in Maple or the tubecurve or Tube command in Mathematica.)