College calc question?

This can actually be done without any equations at all, since the problem is set up to make it easy. Every 14 years, the Sprog/Grak ratio (originally at 8:1) is cut in half because while the Sprogs doubled, the Graks quadrupled. So it'd take three 14-year cycles to do this, or 42 years.

However to set this up on a more general level:

s = 2400 2^(t/14)

g = 300 2^(t/7)

s = g

2400 2^(t/14) = 300 2^(t/7)

8 = 2^(t/7 - t/14)

8 = 2^(t/14)

3 = t/14

t = 42

The problem ultimately doesn't involve calculus at all. It's just a matter of using logarithms.

Calculus not required

2400(2)^(t/14) = 300(2)^(t/7)

8*(2)^(t/14) = (2)^(t/7)

8 = (2)^(t/7) / (2)^(t/14)

8 = 2^(t/7 - t/14)

8 = 2^(t/14)

ln 8 = ln 2^(t/14)

ln 8 / ln 2 = t/14

t = 14* ln 8 / ln 2

t = 42 years

This is how we do it in Ireland;

Divide the 300 Graks by the 2400 sprongs.

The answer (plus a random number) is 200.