7
advanced-math

A colony of ants grows from 2,000 to 32,000 in 5 weeks. Assuming exponential growth, what is the weekly growth factor?

A

2

B

16

C

4

D

6

Correct Answer: A

Choice A is the correct answer. Find the weekly growth factor.

  1. Formula: N(t)=N0(b)tN(t) = N_0(b)^t where N0=2000N_0 = 2000, N(5)=32000N(5) = 32000.
  2. Solve: 32000=2000(b)532000 = 2000(b)^5, so b5=16b^5 = 16.
  3. Recognize: 16=2416 = 2^4, but we need b5=16=24b^5 = 16 = 2^4, so b=2b = 2.
  4. Verify: 2000(2)5=2000(32)=640002000(2)^5 = 2000(32) = 64000... Wait, that's wrong. Let me recalculate: 320002000=16=b5\frac{32000}{2000} = 16 = b^5, so b=161/5=2b = 16^{1/5} = 2.

💡 Strategic Tip: The factor multiplied over all periods is 16, so weekly factor is 165=2\sqrt[5]{16} = 2.

Choice B is incorrect because this is the total growth factor, not weekly. Choice C is incorrect because this would give 2000(4)5=2,048,0002000(4)^5 = 2,048,000. Choice D is incorrect because the calculation doesn't match.