In the year 2000, there were 200,000 cell phone subscribers in a city in New York. The number of subscribers increased by 60 percent per year after 2000. Which equation can be used to model the number of subscribers, $y$, in the city $t$ years after 2000?
A. $y=200,000(1+60)^t$
B. $y=200,000(1+0.6)^t$
C. $y=200,000(1-60)^t$
D. $y=200,000(1-0.6)^t$
Answer (2)
The problem describes an exponential growth scenario where the number of cell phone subscribers increases by a fixed percentage each year.
The general form of the exponential growth equation is y = a ( 1 + r ) t , where a is the initial amount, r is the growth rate, and t is the time.
Substitute the given values into the equation: a = 200 , 000 and r = 0.6 .
The correct equation is y = 200 , 000 ( 1 + 0.6 ) t , so the answer is B .
Explanation
Understanding the Problem We are given that in the year 2000, there were 200,000 cell phone subscribers in a city. The number of subscribers increased by 60 percent per year after 2000. We need to find the equation that models the number of subscribers, y , in the city t years after 2000.
General Exponential Growth Equation The general form of an exponential growth equation is given by: y = a ( 1 + r ) t where:
y is the number of subscribers after t years,
a is the initial number of subscribers,
r is the growth rate (as a decimal),
t is the number of years after 2000.
Substituting the Values In this problem, we have:
Initial number of subscribers, a = 200 , 000
Growth rate, r = 60% = 0.6 Substituting these values into the general equation, we get: y = 200 , 000 ( 1 + 0.6 ) t
Comparing with the Options Now, we compare the equation we derived with the given options: A. y = 200 , 000 ( 1 + 60 ) t B. y = 200 , 000 ( 1 + 0.6 ) t C. y = 200 , 000 ( 1 − 60 ) t D. y = 200 , 000 ( 1 − 0.6 ) t Our derived equation matches option B.
Final Answer Therefore, the correct equation to model the number of subscribers, y , in the city t years after 2000 is: y = 200 , 000 ( 1 + 0.6 ) t
Examples
Exponential growth is a mathematical transformation that increases without bound. A real world example is viral marketing, where more and more people adopt a product or service based on word of mouth. The equation derived in this problem can be used to predict the number of subscribers in the city after a certain number of years, assuming the growth rate remains constant. This can help the city plan for future infrastructure needs and allocate resources accordingly.
The equation that models the number of cell phone subscribers in the city, t years after 2000, is given by y = 200 , 000 ( 1 + 0.6 ) t . Therefore, the correct answer is B. This model reflects a 60% annual growth in subscribers starting from the initial count of 200,000 in the year 2000.
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