Problems IX
- Two electrons repel each other with a force inversely proportional to the square of the distance between them. Suppose that one electron is held fixed at (1,0) on the x-axis. Find the work done to move a second electron along the x-axis from (-1,0) to the origin. Let k = constant of proportionality.
- Find the number which most exceeds its square?
- Two posts, "a" meters apart are respectively "b" meters and "c" meters high. A wire passing through the tops of the posts are held at a point on the ground level between them. Find the location of the point on the ground where the wire is to be pinned to minimize its length?
- If the sum of the areas of a cube and a sphere is constant, what is the ratio of the diameter of the sphere to the side of the cube when the sum of the volumes is a minimum?
- Find the are enclosed by the sine wave (y = sinx) and the cosine wave (y = cosx) between two consecutive points of intersections?
- Computer diskettes contain information in units called bytes that are arranged in concentric circular tracks. Manufacturing constrains limit the density to "x" bytes per centimeter along a given track, and "y" tracks per centimeter measured radially across the disk (where x and y are constants). If the number of bytes on each track must be the same (to achieve uniformity in reading the information), where should the innermost track be located to get the maximum number of bytes on the disk? Let r as the radius of the disk.
- The rate at which the disease AIDS spreads in a certain country is proportional to the product of the number of people infected (HIV positive people) to the number of people not yet infected. The disease is spreading most rapidly when _____?
- Find the area bounded by y2 = 4x and y + 2x = 12?
- Locate the centroid of the plane area bounded by the equation y2 = 4x, x = 1 and the x-axis on the first quadrant?
- Find the length of arc of the parabola x2 = 4y from x = -2 to x = 2?
- The area common to the two parabolas y2 = x and x2 = y is revolved about the y-axis. Find the volume generated?
- The volume of a paraboloid of a revolution is equal to _____?
- Find the area of the largest parabola which can be cut from a right circular cone of radius 3m and an altitude of 4m
- A flower bed is to be in a shape of a circular sector of raius r and central angle q. Find the value of the radius r, if the area is fixed and the perimeter is maximum?
- Find the volume of the largest circular cylinder that can be cut from a right circular cone of radius 2m and height of 3m
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