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Question 15 of 30

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M:C:MC:11

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A biology class at Central High School predicted that a local population of animals will double in size every 12 years. The population at the beginning of 2014 was estimated to be 50 animals. If P represents the population n years after 2014, then which of the following equations represents the class’s model of the population over time?

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P equals 12 plus 50 n

Correct Answer: 
No

P equals 50 plus 12 n

Correct Answer: 
No

P equals 50 open parentheses 2 close parentheses to the power of 12 n end exponent

Correct Answer: 
No

P equals 50 open parentheses 2 close parentheses to the power of n over 12 end exponent

Correct Answer: 
Yes

View Correct Answer

Choice D is correct. The student first recognizes that a population that doubles in size over equal time periods is increasing at an exponential rate. An exponential growth model can be written in the form y equals a open parentheses 2 close parentheses to the power of x over b end exponentwhere a is the population at time 0, since 2 raised to the zeroth power is 1 and b is the doubling time, since when x equals 12 comma 2 is raised to the first power, and the population will be 2 to the power of 1 cross times a equals 2 a. From the way the variables were defined, the population at time n equals 0 is 50 and the doubling time is 12.

Choice A is not the correct answer. This answer may result from a misunderstanding of exponential equations or of the context. A student who tries to model the scenario with a linear equation or who misunderstands that the y-intercept of a linear model should represent the initial number of animals may produce this equation.

Choice B is not the correct answer. This answer may result from a misunderstanding of exponential equations or of the scenario. A student who tries to model the scenario with a linear equation may produce this equation.

Choice C is not the correct answer. A student who tries to produce an exponential model, but does not understand how the 12 years affects the model, may incorrectly write the exponent.

Question Difficulty: 
medium
Objective: 

Students must identify the correct mathematical notation for an exponential relationship that represents a real-world situation.