(a) The cost of M gallons of medicine and N gallons of nutrients is 25M+5N. Note that, to save the greatest number of mammals, your team should spend the entire $1,000 on nutrients and medicine since the habitat is too large to overdose, and since your model for how many lives are saved, s = M²+NM+12N is a function which gets bigger if either M or N are increased. Hence, if you spent less than $1,000, this couldn't cause the greatest number of mammals to be saved, since then spending the extra funds you have on either medicine or nutrients would make this function (number of lives saved)bigger. (b) Hence, whatever M and N need to be to save the greatest number of lives, you will have 25M+5N = 1000

(c) To maximize the number of mammals saved, we first use the last equation to eliminate one of the two variables from the formula for s, so that s becomes a function of one variable, which your team knows how to maximize from the single-variable algebra course you all took. To avoid fractions, you might as well solve for N (rather than M), getting 5N = 1000-25M, or N = 200-5M.

(c) Substituting this into N in your formula for s gives M² + (200-5M)M + 12(200-5M), an expression s(M) only in M for the total number of lives saved by next year. Expanding, this becomes M² + 200M - 5M² + 2400 - 60M, which simplifies to -4M² + 140M + 2400.

To find the maximum, we "complete the square":

Thus the quadratic function (parabola) -4M² + 140M + 2400 can be re-written as -4(M - (17.5))² + 3625 whose vertex is clearly (17.5, 3625).

Since this parabola gives the total number of lives saved, we see that to maximize the number of lives saved, we should let M=17.5 (gallons of medicine), and that this choice would result in 3,625 adorable marine mammals saved.