29. The amount of coffee consumed annually in Country X increases by 8 percent each year. If 970 thousand metric tons of coffee were consumed in 2000, which of the following functions c can be used to model the amount of coffee, in thousands of metric tons, consumed annually in Country X, where t is the number of years since 2000?
Answer: C
c(t) = 970(1.08)^t
The function that accurately models the annual coffee consumption in Country X, given an 8 percent increase each year, is c(t) = 970(1.08)^t. This equation represents exponential growth, which is appropriate for scenarios where a quantity increases by a fixed percentage over time.
A) c(t) = 970 + 1.08t
This option represents a linear function, indicating a constant increase in coffee consumption each year. However, since the consumption grows by a percentage, not a fixed amount, this model does not accurately reflect the compounding nature of the annual increase.
B) c(t) = 970 (1.08t)
While this option appears to incorporate the initial consumption and a growth factor, it incorrectly uses a linear exponent with t. This form does not represent exponential growth, as it does not account for the compounding effect of the 8 percent increase over each year.
C) c(t) = 970(1.08)^t
This is the correct option as it models exponential growth, where 970 is the initial consumption in thousands of metric tons, and (1.08)^t accounts for the 8 percent increase each year compounded over time. This function correctly captures the growth pattern described.
D) c(t) = 970^(1.08) t
This option incorrectly suggests that the initial consumption is raised to the power of 1.08, which misrepresents the relationship between time and the growth factor. It does not represent either linear or exponential growth accurately.
E) c(t) = 970 t^(1.08)
This function incorrectly implies that coffee consumption increases with time raised to the power of 1.08, suggesting a non-linear growth model that does not match the scenario of a percentage increase over time. This does not accurately represent the compounding nature of the growth.
Conclusion
The correct function c(t) = 970(1.08)^t effectively models the scenario of coffee consumption increasing by 8 percent annually by applying an exponential growth model. All other options fail to capture the correct relationship of compounded growth, either presenting linear growth or misapplying the mathematical principles related to percentage increases.