Bubbles Imagine a stack of hemispherical soap bubbles with decreasing radii r, = 1, r, ľ3, ... (see figure). Let h, be the dis- tance between the diameters of bubble n and bubble n + 1, and let H, be the total height of the stack with a bubbles. a. Use the Pythagorean theorem to show that in a stack with n bubbles, h = r - rž, h = r - r, and so forth. Note that for the last bubble h, = r b. Use part (a) to show that the height of a stack with a bubbles is H, = VrR - rị + VrR - r; + · + Vr-1 - r, + r c. The height of a stack of bubbles depends on how the radii de- crease. Suppose r, = 1, r, = a, r; = a², . ,r, = a²-!, where 0 < a < iis a fixed real number. In terms of a, find the height H, of a stack with n bubbles. d. Suppose the stack in part (c) is extended indefinitely (n → ). In terms of a, how high would the stack be? e. Challenge problem: Fix n and determine the sequence of radii r1, 12, r3, ...,r, that maximizes H, the height of the stack with n bubbles.