3
advanced-math

A city's renewable energy adoption follows E(t)=1001+19e0.4tE(t) = \frac{100}{1 + 19e^{-0.4t}} percent where tt is years. What is the long-term adoption limit?

A

100%

B

50%

C

5%

D

19%

Correct Answer: A

Choice A is the correct answer. Find the carrying capacity.

  1. As tt \to \infty: e0.4t0e^{-0.4t} \to 0.
  2. Limit: E()=1001+0=100%E(\infty) = \frac{100}{1 + 0} = 100\%.
  3. Result: Maximum adoption is 100%.

💡 Strategic Tip: In K1+Aert\frac{K}{1 + Ae^{-rt}}, the numerator KK is always the limit.

Choice B is incorrect because 50% would be if K=50K = 50. Choice C is incorrect because this is the initial adoption. Choice D is incorrect because 19 is the coefficient AA, not the limit.