Set 15: Exponential Functions

Explanation

Answer: B

A startup's user base is increasing by 12% every month. If the initial user base is $P_0$ , which function models the number of users $U(t)$ after $t$ months?

$U(t) = P_0(0.12)^t$

$U(t) = P_0(1.12)^t$

✓ Correct

$U(t) = P_0(12)^t$

$U(t) = P_0(0.88)^t$

Detailed Explanation

Choice B is correct. Choice B is the correct answer. Growth rate $r$ converts to growth factor $b = 1 + r$ . 1. Identify Rate: The rate of increase is $r = 12\% = 0.12$ . 2. Calculate Factor: For growth, $b = 1 + r = 1 + 0.12 = 1.12$ . 3. Form Equation: Substitute into $y = ab^x$ to get $U(t) = P_0(1.12)^t$ . Strategic Tip: "Increasing by $r\%$ " always means multiplying by $1+ r$ (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% ( $1- 0.12$ ).

Key Steps:

•

The correct answer is $U(t) = P_0(1.12)^t$

Why others are wrong:

A: Choice A is incorrect and may result from a calculation error.

C: Choice C is incorrect and may result from a calculation error.

D: Choice D is incorrect and may result from a calculation error.

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Set 15: Exponential Functions

Detailed Explanation

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