Question
An engineer collects data on power (in kilowatts) generated by a wind turbine over time. At time t=16 weeks, the turbine’s production is 91 kW, and at time t=18 weeks, the turbine’s production is 150 kW. Find the average rate of change from t=16 to t=18 weeks, rounding your answer to three decimal places, with appropriate units. Then fill in the guided sentence below to interpret the meaning of your answer in context.
Answer
The average rate of change is
. It represents that the
is
at a rate of
over that time interval.
Answer
The average rate of change is
. It represents that the
is
at a rate of
over that time interval.
Answers
GPT-4o mini
To find the average rate of change of the turbine's power production from \( t = 16 \) weeks to \( t = 18 \) weeks, we can use the formula for average rate of change:
\[
\text{Average Rate of Change} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}
\]
In this case, let \( f(t) \) represent the power production at time \( t \). We have:
- \( f(16) = 91 \) kW (at \( t = 16 \) weeks)
- \( f(18) = 150 \) kW (at \( t = 18 \) weeks)
Substituting the values into the formula:
\[
\text{Average Rate of Change} = \frac{150 - 91}{18 - 16} = \frac{59}{2} = 29.5
\]
The average rate of change is \( 29.5 \) kW per week.
Now, we can fill in the guided sentence:
The average rate of change is **29.5 kW per week**. It represents that the **power production of the wind turbine** is **increasing** at a rate of **29.5 kW per week** over that time interval.
\[
\text{Average Rate of Change} = \frac{f(t_2) - f(t_1)}{t_2 - t_1}
\]
In this case, let \( f(t) \) represent the power production at time \( t \). We have:
- \( f(16) = 91 \) kW (at \( t = 16 \) weeks)
- \( f(18) = 150 \) kW (at \( t = 18 \) weeks)
Substituting the values into the formula:
\[
\text{Average Rate of Change} = \frac{150 - 91}{18 - 16} = \frac{59}{2} = 29.5
\]
The average rate of change is \( 29.5 \) kW per week.
Now, we can fill in the guided sentence:
The average rate of change is **29.5 kW per week**. It represents that the **power production of the wind turbine** is **increasing** at a rate of **29.5 kW per week** over that time interval.