The function $f(x)=68(1.3)^x$ represents the possible squirrel population in ark $x$ years from now. Each year, the expected number of squirrels is $\qquad$ umber the year before.
A. 3 more than
B. 1.3 times
C. 3 times
D. 0.3 times
Answer (2)
The function f ( x ) = 68 ( 1.3 ) x models the squirrel population.
Calculate the ratio of the population in year x + 1 to year x : f ( x ) f ( x + 1 ) = 68 ( 1.3 ) x 68 ( 1.3 ) x + 1 .
Simplify the ratio: 68 ( 1.3 ) x 68 ( 1.3 ) x + 1 = 1.3 .
Each year, the expected number of squirrels is 1.3 times the number the year before. The answer is 1.3 times .
Explanation
Understanding the Function The function given is f ( x ) = 68 ( 1.3 ) x . This function models the squirrel population x years from now. We want to determine how the population changes each year.
Comparing Consecutive Years To find out how the population changes from one year to the next, we can compare the population in year x + 1 to the population in year x . That is, we want to find the ratio f ( x ) f ( x + 1 ) .
Calculating the Ratio We have f ( x + 1 ) = 68 ( 1.3 ) x + 1 . So, the ratio is: f ( x ) f ( x + 1 ) = 68 ( 1.3 ) x 68 ( 1.3 ) x + 1 .
Simplifying the Ratio We can simplify this ratio by canceling out the common factors: 68 ( 1.3 ) x 68 ( 1.3 ) x + 1 = ( 1.3 ) x ( 1.3 ) x + 1 = 1. 3 x + 1 − x = 1. 3 1 = 1.3 This means that the population in year x + 1 is 1.3 times the population in year x .
Conclusion Therefore, each year, the expected number of squirrels is 1.3 times the number the year before.
Examples
Exponential functions like the one in this problem are used to model population growth in various scenarios, such as bacteria in a petri dish, or the spread of a virus. Understanding how to analyze these functions helps us predict future population sizes and make informed decisions about resource management or public health interventions. For example, if we know the growth rate of a bacterial colony, we can predict how quickly it will reach a certain size and determine the appropriate amount of antibiotic to use.
The expected number of squirrels each year is 1.3 times greater than the previous year's population, as derived from the function f ( x ) = 68 ( 1.3 ) x . So, the answer to the question is 1.3 times .
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