The population of the world was about 6.1 billion in

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 […]

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(a) Set up, but do not evaluate, a double integral

(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 […]

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A solid occupies a region E with surface S and

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 […]

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Suppose the three coordinate planes are all mirrored and a

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 […]

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The magnitude of a velocity vector is called speed. Suppose

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 […]

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A molecule of methane, CH4, is structured with the four

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. […]

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A solid has the following properties. When illuminated by rays

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 […]

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The view of the trefoil knot shown in Figure 8

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 ty = (2 + cos 1.5t) sin tz = sin 1.5t to sketch the curve by hand as viewed from above, with gaps indicating where […]

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