a) We are going to complete the square for the equation y=0.2t2-6.4t+12.
Factorize 0.2 out of the first two terms, y=0.2(t2-32t)+12
Now, complete the square by adding (32/2)2=162=256 inside of the parenthesis AND subtracting 256x0.2, which equals 51.2. We have to subtract 51.2 to keep the value the same (to see why we don't just subtract 256, use distribution): y=0.2(t2-32t+256)+12-51.2
This simplifies to y=0.2(t-16)2-39.2. Now, we can see that the minimum value of y is -39.2, by making t equal to 16.
To see why this is the minimum, look at (t-16)2. The square of any real number > 0. So the minimum value of (t-16)2 is 0.
Therefore, the lowest depth Rosy dived is 39.2 meters.
**My solution is a bit hard to understand if you've never worked with Completing the Square or minimizing quadratics. Most Algebra 2 books cover these topics.