5
advanced-math

A startup's user base is increasing by 12% every month. If the initial user base is P0P_0, which function models the number of users U(t)U(t) after tt months?

A

U(t)=P0(0.12)tU(t) = P_0(0.12)^t

B

U(t)=P0(1.12)tU(t) = P_0(1.12)^t

C

U(t)=P0(12)tU(t) = P_0(12)^t

D

U(t)=P0(0.88)tU(t) = P_0(0.88)^t

Correct Answer: B

Choice B is the correct answer. Growth rate rr converts to growth factor b=1+rb = 1 + r.

  1. Identify Rate: The rate of increase is r=12%=0.12r = 12\% = 0.12.
  2. Calculate Factor: For growth, b=1+r=1+0.12=1.12b = 1 + r = 1 + 0.12 = 1.12.
  3. Form Equation: Substitute into y=abxy = ab^x to get U(t)=P0(1.12)tU(t) = P_0(1.12)^t.

💡 Strategic Tip: "Increasing by r%r\%" always means multiplying by 1+r1 + r (as a decimal).

Choice A is incorrect because 0.12 would mean the population shrinks to 12% of its size each month. Choice C is incorrect because 12 would mean growing by 1100% (multiplying by 12). Choice D is incorrect because 0.88 represents a decrease of 12% (10.121 - 0.12).