21:56

Calculus for Scientists and Engineers: Early Transcendental

Bubbles Imagine a stack of hemispherical soap bubbles with decreasing radii $r_{1}=1, r_{2}, r_{3}, \ldots$ (see figure). Let $h_{n}$ be the distance between the diameters of bubble $n$ and bubble $n+1,$ and let $H_{n}$ be the total height of the stack with $n$ bubbles. a. Use the Pythagorean theorem to show that in a stack with $n$ bubbles, $h_{1}^{2}=r_{1}^{2}-r_{2}^{2}, h_{2}^{2}=r_{2}^{2}-r_{3}^{2},$ and so forth. Note that $h_{n}=r_{n}$

b. Use part (a) to show that the height of a stack with $n$ bubbles is

$$H_{n}=\sqrt{r_{1}^{2}-r_{2}^{2}}+\sqrt{r_{2}^{2}-r_{3}^{2}}+\cdots+\sqrt{r_{n-1}^{2}-r_{n}^{2}}+r_{n}$$

c. The height of a stack of bubbles depends on how the radii decrease. Suppose that $r_{1}=1, r_{2}=a, r_{3}=a^{2}, \ldots, r_{n}=a^{n-1}$

where $0<a<1$ is a fixed real number. In terms of $a$, find the height $H_{n}$ of a stack with $n$ bubbles.

d. Suppose the stack in part (c) is extended indefinitely $(n \rightarrow \infty)$ In terms of $a$, how high would the stack be?

e. Challenge problem: Fix $n$ and determine the sequence of radii $r_{1}, r_{2}, r_{3}, \ldots, r_{n}$ that maximizes $H_{n},$ the height of the stack with $n$ bubbles. (FIGURE CANNOT COPY)