Chapter 4

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?

An alternative method by which Archimedes could have solved the crown problem is given by Vitruvius in \(\mathrm{On}\) Architecture. Assume as in the text that the crown is of weight \(W\), composed of weights \(w_{1}\) and \(w_{2}\) of gold and silver, respectively. Assume that the crown displaces a certain quantity of fluid, \(V\). Furthermore, suppose that a weight \(W\) of gold displaces a volume \(V_{1}\) of fluid, while a weight \(W\) of silver displaces a volume \(V_{2}\) of fluid. Show that \(V=\frac{w_{1}}{W} V_{1}+\frac{w_{2}}{W} V_{2}\) and therefore that \(\frac{w_{1}}{w_{2}}=\frac{V_{2}-V_{1}}{V-V_{1}}\). Fig. 4.7)

Use calculus to prove Archimedes' result from The Method that the volume of the segment of the cylinder described in the text is \(1 / 6\) the volume of the rectangular parallelepiped circumscribing the cylinder.

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 intersect in at most four points.

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

Prove analytically Proposition VII-12, that in any ellipse the sum of the squares on any two of its conjugate diameters is equal to the sum of the squares on its two axes. (In Figure \(4.32\), this means that \(P G^{2}+D K^{2}=A E^{2}+B L^{2}\) )

Use Proposition II-8 to show that the two line segments of a tangent to a hyperbola between the point of tangency and the asymptotes are equal. Then show, without calculus, that the slope of the tangent line to the curve \(y=1 / x\) at \(\left(x_{0}, 1 / x_{0}\right)\) equals \(-1 / x_{0}^{2}\)

Prove Proprositicet III-45 foe an ellipse, namely, if \(A C\) aad B \(D \mathrm{~ a r e ~ t a n g e n t ~ w o ~ l h e ~ e l l i p o e ~ a t ~ u h e ~ t w a ~ e f}\) axis, atd af \(C D\) is taagent wo the ellipse at \(E_{1}\) and if oene conacets \(C, D\), to the swo foci \(F, C\), respectively, then angles \(C F D\) and \(D V D C\) ast tigh angles (Fig- 4.34) (See section 4.5.3 foe the defititich of foci.) FLCURE 434 $\mathrm{~

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