The design of an electric power generating station depends on both the peak and the average power that it must produce. If a community uses 336 plus 144 t minus 6 t squared336+144t−6t2 megawatts at time t​ (in hours) during the period t equals 0t=0 to t equals 24t=24​, find the average level of power consumption for that day

To find the average level of power consumption for the given power function over a day, we need to calculate the average value of the function \( P(t) = 336 + 144t - 6t^2 \) over the interval from \( t = 0 \) to \( t = 24 \).

The average value of a continuous function \( f(t) \) over an interval \([a, b]\) is given by:

\[
\text{Average value} = \frac{1}{b-a} \int_a^b f(t) \, dt
\]

Here, \( f(t) = P(t) = 336 + 144t - 6t^2 \), \( a = 0 \), and \( b = 24 \). So, we need to compute:

\[
\text{Average power consumption} = \frac{1}{24 - 0} \int_0^{24} (336 + 144t - 6t^2) \, dt
\]

First, we find the definite integral:

\[
\int_0^{24} (336 + 144t - 6t^2) \, dt
\]

We integrate each term separately:

\[
\int_0^{24} 336 \, dt + \int_0^{24} 144t \, dt - \int_0^{24} 6t^2 \, dt
\]

\[
= 336 \left[ t \right]_0^{24} + 144 \left[ \frac{t^2}{2} \right]_0^{24} - 6 \left[ \frac{t^3}{3} \right]_0^{24}
\]

\[
= 336 \left[ t \right]_0^{24} + 144 \left[ \frac{t^2}{2} \right]_0^{24} - 6 \left[ \frac{t^3}{3} \right]_0^{24}
\]

Evaluating each term at the bounds \( t = 0 \) and \( t = 24 \):

\[
336 \left[ 24 - 0 \right] + 144 \left[ \frac{24^2}{2} - \frac{0^2}{2} \right] - 6 \left[ \frac{24^3}{3} - \frac{0^3}{3} \right]
\]

\[
= 336 \cdot 24 + 144 \cdot \frac{576}{2} - 6 \cdot \frac{13824}{3}
\]

\[
= 8064 + 144 \cdot 288 - 6 \cdot 4608
\]

\[
= 8064 + 41472 - 27648
\]

\[
= 21888
\]

Now, we divide by the length of the interval \( b - a = 24 - 0 = 24 \):

\[
\text{Average power consumption} = \frac{21888}{24} = 912 \text{ megawatts}
\]

So, the average level of power consumption for the community over the day is **912 megawatts**.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How do you derive the formula for the average value of a function?
2. Can the average power consumption be different if we change the time interval?
3. What are the units of the integral of a power function with respect to time?
4. How does the quadratic term in the power function affect the total power consumption over the day?
5. What is the significance of calculating the average power consumption for an electric power station?
6. How would the result change if the power function had a cubic term instead?
7. How do you calculate the definite integral of a polynomial function?
8. What methods can be used to verify the correctness of the integral computation?

**Tip:** When working with integrals of polynomial functions, breaking them down term by term simplifies the integration process and helps avoid mistakes.

Learn how to calculate the average power consumption of a community over a day using integral calculus. This problem involves finding the average value of a polynomial function over a specific time interval. Suitable for advanced high school and college students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation