Exponential writing assignment

Question

Lex "JC" Luthor is tired of his long-time fiend Superman. JC has decided that he is going to take him out with kryptonite, so JC has been using his money to grow the awesome green material JC knows that to destroy Superman he will need to have at least 300 tons of kryptonite . He started with 50 tons of kryptonite, and it is growing at a continuous rate. He knows that in 3 weeks, he will have 100 tons of kryptonite

1-What is the growth rate (round to 3 decimals)?

2-How long will JC have to wait until he has enough to take out Superman (round to 3 decimals)?

Write about the key features (domain, range, growth/decay, y-intercept, end behavior, asymptotes, and increasing/decreasing behavior) and how you found them and or came this conclusion

Don't forget to answer the two questions and how you calculated those answers

Answer 1

The general exponential equation can be written as y = ab^x where be is the growth rate and a is the initial amount, x is the time and y is the future value. So, in this case, 100 = 50(b)³ Note that it is critical that the growth rate and the units of time match each other. So, since we used 3 weeks in the exponent, we will be calculating "growth rate per week".

100 = 50(b)³

100/50 = b³

b = cuberoot(2) = 1.25992 or, in other words, the growth rate is 25.992% per week (growth because the number was larger than 1 and 25.992% because it is 0.25992 more than 1)

Since he needs 300 tons, we can calculate t by plugging in the 300 like this:

300 = 50(1.25992)^t

300/50 = (1.25992)^t

6 = 1.25992^t

log_1.25992(6) = t = 7.755 weeks

The domain of the function y = 50(1.25992)^t is t > 0 (this is a logical domain not a mathematical one. It makes sense that time will be greater than 0 because he is starting with 50 tons and can't go backwards in time (negative t). The range is y > 50 tons because, again, he is starting with 50 tons and it is growing. This is a growth rate because the amount of kryptonite is growing.The y intercept is when t = 0, so (0, 50) and the end behavior is that as t →∞, y →∞ (y grows without bound). Mathematically, there is an asymptote at y = 0 but this is not practical in this particular problem because we can't go backwards in time to approach the asymptotic limit. This function is always increasing over its domain.