China's push to install more wind energy capacity has...

China's push to install more wind energy capacity has started paying off. As of 2012 , wind energy had become the country's third largest source of energy, after coal and hydroelectric power. The amount of wind energy generated in China from 2005 through 2012 grew at the rate of $$r(t)=5.018 t-3.204 \quad(0 \leq t \leq 7)$$ terawatt-hours/year, in year $t,$ where $t=0$ corresponds to $2005 .$ The amount of wind energy generated in 2005 stood at 1.8 terawatt-hours. a. Find an expression giving the amount of wind energy generated in year $t$ b. How much wind energy was generated in $2012 ?$ c. If the trend continued into 2013 , how much wind energy was generated in China in that year?

China's push to install more wind energy capacity has started paying off. As of 2012 , wind energy had become the country's third largest source of energy, after coal and hydroelectric power. The amount of wind energy generated in China from 2005 through 2012 grew at the rate of
$$r(t)=5.018 t-3.204 \quad(0 \leq t \leq 7)$$
terawatt-hours/year, in year $t,$ where $t=0$ corresponds to $2005 .$ The amount of wind energy generated in 2005 stood at 1.8 terawatt-hours.
a. Find an expression giving the amount of wind energy generated in year $t$
b. How much wind energy was generated in $2012 ?$
c. If the trend continued into 2013 , how much wind energy was generated in China in that year?

Key Concepts

Rate Function

A rate function represents the instantaneous change of a quantity with respect to time. In the context of this problem, it indicates how much the production of wind energy changes per year. Grasping this concept is critical, as it forms the basis for connecting the rate of change to the overall amount produced through integration.

Definite Integration

Definite integration is a mathematical process used to calculate the accumulation of a quantity over a specific interval, when the rate of change is known. By integrating the rate function, one can derive a function that represents the total amount of the quantity—in this case, wind energy—accumulated from a certain starting point in time.

Initial Value Problem

An initial value problem involves finding a specific solution to a differential equation that satisfies an initial condition. Here, the known value of wind energy production at the starting year is used to determine the constant of integration, ensuring that the specific solution matches the physical situation described.

Extrapolation of Models

Extrapolation involves extending a mathematical model beyond the interval for which it was originally defined or validated. In this problem, the model for the rate of wind energy generation is applied to predict production in a year beyond the data range, highlighting the importance of understanding the limitations and assumptions of the modeled scenario.

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