91Ó°ÊÓ

Find where to place the fulcrum in a lever of length \(10 \mathrm{~m}\) so that a weight of \(14 \mathrm{~kg}\) at one end will balance a weight of \(10 \mathrm{~kg}\) at the other.

If a weight of \(8 \mathrm{~kg}\) is placed \(10 \mathrm{~m}\) from the fulcrum of a lever and a weight of \(12 \mathrm{~kg}\) is placed \(8 \mathrm{~m}\) from the fulcrum in the opposite direction, toward which weight will the lever incline?

Use calculus to prove Archimedes' result that the area of a parabolic segment is four-thirds of the area of the inscribed triangle.

Use calculus to prove Archimedes' result that a cylinder whose base is a great circle in the sphere and whose height is equal to the diameter of the sphere has volume \(3 / 2\) that of the sphere and also has surface area \(3 / 2\) of the surface area of the sphere.

Use calculus to prove Archimedes' result that the area bounded by one complete turn of the spiral given in poIar coordinates by \(r=a \theta\) is one-third of the area of the circle with radius \(2 \pi a\).

Consider Proposition 1 of \(O n\) the Sphere and Cylinder II: Given a cylinder, to find a sphere equal to the cylinder. Provide the analysis of this problem. That is, assume that \(V\) is the given cylinder and that a new cylinder \(P\) has been constructed of volume \(\frac{3}{2} V .\) Assume further that another cylinder \(Q\) has been constructed equal to \(P\) but with height equal to its diameter. The sphere whose diameter equals the height of \(Q\) would then solve the problem, because the volume of the sphere is \(\frac{2}{3}\) that of the cylinder. So given the cylinder \(P\) of given diameter and height, determine how to construct a cylinder \(Q\) of the same volume but whose height and diameter are equal.

Show that in the curve \(y^{2}=p x\), the value \(p\) represents the length of the latus rectum, the straight line through the focus perpendicular to the axis.

Demonstrate analytically Apollonius's result from Book IV that two conic sections can be tangent at no more than two points.

Can one consider Archimedes as an inventor of the integral calculus?