Calculus two question

Question

In a recent archaeological expedition, a scroll was discovered containing a description of a plan to build what appeared to be the tower babel. According to the manuscript, the tower was supposed to have a circular cross-section and "go up to the heavens" (infinitely high). A mathematician was consulted to answer some of the questions posed by the archeologists. The mathematician plotted half of the silhouette of the tower on a set of coordinate axes with the y-axis passing through the center, and discovered that is was approximated by the curve y = (-100l)n(/5).

a.) calculate the volume of the tower

b.) at that time, babel's unit of length was called a shrim. The manuscript mentions that 4200 cubic shrim of stone were available to build the tower. Based on your answer above, was there enough material on hand to complete the tower?

Answer 1

While it's not entirely clear what equation was intended for the tower's cross-section, a reasonable guess would be y = -100 ln(x/5). Since we're told that the tower has a circular cross-section, the full 3D shape of the tower is obtained by revolving the given silhouette around the y-axis. The volume will then be the sum (that is, the integral) of these circular cross-sections, so all that remains is to find the infinitesimal volume dV of the cross-section at height y.

To do this, we note that each cross-section can be accurately approximated as an infinitesimally thin cylinder with height dy. The base area of each cylinder is of the form πr², where r is the distance from the axis of rotation (in this case, the y-axis) to a point on the curve (in this case, y = -100 ln(x/5)). Since this distance is purely horizontal in this case, we have r = x, where x is the x-coordinate of the point with vertical coordinate y on our given curve. This means that the infinitesimal volume of one cylindrical cross-section is given by dV = πx²dy. Since x depends on y, we need to express x in terms of y in order to integrate this infinitesimal volume. To do so, we solve y = -100 ln(x/5) for x:

-y/100 = ln(x/5)

e^-y/100 = x/5

5e^-y/100 = x

We have x² in our integrand, so what we're really interested in is x² = 25e^-y/50, which we obtain by squaring both sides of the above equation. Therefore, our infinitesimal volume becomes dV = 25πe^-y/50dy, which we can now integrate.

The tower stretches infinitely high, so the upper limit of our integral is +∞. Assuming that the ground is at y = 0, we therefore want to calculate ∫₀^∞ 25πe^-y/50dy, which is equal to

25π ∫₀^∞ e^-y/50dy

= 25π [-50e^-y/50]₀^∞

= -1250π (0 - 1)

= 1250π

≈ 3927

This is less than the total amount of stone available, so they would have been able to construct the full tower with the materials available.