In a recent archaeological expedition, a scroll was discovered containing a description of a plan to build what appears to be the Tower of Babel.According to the manuscript, the tower was supposed to have a circular cross section and "go up to the heavens" (i.e., be infinitely high).A mathematician was consulted to solve some of the questions posed by the archaeologists.The mathematicians plotted half of the silhouette of the tower on a set of coordinate axes with the y-axis running through the center and discovered that it was approximated by the curve $y=-100 \ln \left(\frac{x}{4}\right)$ .The manuscript mentions that 2560 cubic "shrims" (Babel's unit of length)of stones were available to build the tower.The base of the tower was to have radius 3 shrims.Did they have enough?

Question

Quizplus · Accepted Answer

The volume of the tower is given by the integral of the area of the cross-section from $x=0$ to $x=\infty$. The area of the cross-section is given by $\pi y^2$, so the volume is$$V=\int_0^\infty \pi y^2 dx = \int_0^\infty \pi (-100 \ln \left(\frac{x}{4}\right))^2 dx$$This integral diverges, which means that the tower would have infinite volume and therefore could not be built with a finite amount of material.

In a Recent Archaeological Expedition, a Scroll Was Discovered Containing y=−100ln⁡(x4)y=-100 \ln \left(\frac{x}{4}\right)y=−100ln(4x​)

In a Recent Archaeological Expedition, a Scroll Was Discovered Containing $y=-100 \ln \left(\frac{x}{4}\right)$