Question

A hydraulic engineer collects data on hydroelectric production (in megawatts) from a dam over time. At time t=2 weeks, the dam’s production is 1391 MW, and at time t=10 weeks, the dam’s production is 1882 MW. Find the average rate of change from t=2 to t=10 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.
The average rate of change is

. It represents that the
is
at a rate of

over that time interval.

Answer 1

To find the average rate of change from \( t = 2 \) weeks to \( t = 10 \) weeks, we use the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where:
- \( f(a) \) is the production at time \( t = a \)
- \( f(b) \) is the production at time \( t = b \)
- \( a = 2 \) weeks
- \( b = 10 \) weeks
- \( f(2) = 1391 \, \text{MW} \)
- \( f(10) = 1882 \, \text{MW} \)

Plugging in the values:

\[
\text{Average Rate of Change} = \frac{1882 \, \text{MW} - 1391 \, \text{MW}}{10 \, \text{weeks} - 2 \, \text{weeks}}
\]

Calculating the numerator and denominator:

\[
= \frac{491 \, \text{MW}}{8 \, \text{weeks}} = 61.375 \, \text{MW/week}
\]

Now rounding to three decimal places gives:

\[
\text{Average Rate of Change} = 61.375 \, \text{MW/week}
\]

Now, we can fill in the guided sentence:

The average rate of change is **61.375 MW/week**. It represents that the **hydroelectric production** is **increasing** at a rate of **61.375 MW** over that time interval.