Fish and squares.

Currently I am working on two different problems, both of which I have added to the Current and Future projects page.

To summarize, these are…

1. A Fish Population Problem started by Gaurav Tiwari and his blog “My Digital Notebook”. A link to the post can be found here. A summary of the problem is as follows:

Let $F_{m}(t)$ and $F_{f}(t)$ denote the number of male and female fish, respectively, inside a tank at time t (where t is in a unit of days) and let $F(t)=F_{m}(t)+F_{f}(t)$ denote the total number of fish at time t. Initially we have $F_{m}(0)=4$ and $F_{f}(0)=3$.

Here are the conditions of the problem: A male fish can mate with any number of female fish, unless the female has already mated before. Female fish will always give birth to new offspring after 21 days of conception, with the first female always giving birth to 100 eggs, the second fish, 110 eggs, and the third, 90 eggs.

The new offspring fish will reach sexually maturity after 30 days and in each new batch, 60% of the offspring fish will die before reaching maturity. In the remaining 40%, the ratio of male to female fish is 3:2.

This second generation of fish have the following restrictions. Fish born of the same mother may note mate with each other and as with the first generation, a male fish can mate with any number of female fish, unless the female has already mated before. Every three females who give birth will always lay the eggs in the following order (from first fish to last): 100, 110, and 90. In these batches of eggs, only 40% will survive, with the new ratio of male to female being 1:1. In subsequent generations, the same restrictions found in the second generation of offspring will also apply.

Find the value of $E\left[F(365)\right]$, that is, the expected total population of fish found after one year has passed.”

2. A geometry problem that was started by a friend of a friend taking a Calculus III course at the University of  McMaster. Currently I am trying to find:

…how to calculate the volume of the solid displaced when a square plate of side length 2 is symmetrically passed through a sphere or radius 2 (the center of the square is threaded through a diameter line in the sphere).

as well as trying to attempt to solve the integral that represents said volume.

Today I have made considerable progress in both problems (for 1, see here, and for 2, see  here), but now I am at an impasse. This is why I would like to ask you, the readers, to give me some inspiration and ideas here but helping me answer the following questions:

1. For project 1, what suitable restrictions should be made so that the population can naturally be imbued with a carrying capacity?

2. For project 2, what would be a simple and intuitive way to integrate the $\arcsin$ portion of the integral.

3. For project 2, is there any way to re-write the integral to make the integration simpler?