12. 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 Country X 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 models the amount of coffee consumed annually in Country X is c(t) = 970 * (1.08) ^ t, which represents exponential growth due to the 8 percent increase each year.
A) c(t) = 970 + 1.08t
This option represents a linear function, which suggests that the coffee consumption increases by a constant amount each year (1.08 thousand metric tons). However, the question states that the consumption increases by a percentage each year, indicating exponential growth rather than a fixed increase.
B) c(t) = 970(1.08t)
This option incorrectly applies the exponent. While it includes the initial consumption of 970 thousand metric tons, it incorrectly implies that the growth rate is based on t as a whole rather than compounding annually, leading to incorrect calculations over time.
C) c(t) = 970 * (1.08) ^ t
This option correctly models the coffee consumption using an exponential function, where 970 is the initial amount consumed in 2000 and 1.08 represents the annual growth factor. As t increases, the consumption grows exponentially at an 8 percent rate, aligning with the information provided.
D) c(t) = 970 ^ 1.08 * t
This option suggests a multiplicative relationship but misapplies the parameters by raising the initial consumption to the power of 1.08 and multiplying by t. It does not represent the annual compounding percentage increase and thus fails to model the situation correctly.
E) c(t) = 970t ^ 1.08
This function implies that the consumption is proportional to t raised to the power of 1.08, which indicates polynomial growth rather than the desired exponential growth. This does not reflect the annual percentage increase in consumption.
Conclusion
The correct model, c(t) = 970 * (1.08) ^ t, effectively represents the annual 8 percent increase in coffee consumption, capturing the essence of exponential growth. All other options either misinterpret the nature of the growth (linear or polynomial) or incorrectly apply the formula, leading to inaccurate models. Thus, Option C is the only valid choice based on the given context.