worldwide wind generating capacity W was 371.4 gigawatts at the end of 2014 and 591.1 gigawatts at the end of 2018 (t is years since 2014) write W as an exponential function of T



To model the worldwide wind generating capacity \( W \) as an exponential function of time \( t \), where \( t \) represents the number of years since 2014, we can use the general form of an exponential function:

\[
W(t) = W_0 \cdot e^{kt}
\]

Where:
- \( W_0 \) is the initial value of \( W \) (capacity at \( t = 0 \), which corresponds to 2014).
- \( k \) is the growth rate.
- \( t \) is the time in years since 2014.

Given:
- In 2014 (\( t = 0 \)), the capacity was \( W_0 = 371.4 \) gigawatts.
- In 2018 (\( t = 4 \)), the capacity was \( W(4) = 591.1 \) gigawatts.

### Step 1: Substitute known values into the exponential equation for \( t = 4 \):

\[
591.1 = 371.4 \cdot e^{4k}
\]

### Step 2: Solve for \( k \):

Divide both sides by 371.4:

\[
\frac{591.1}{371.4} = e^{4k}
\]

\[
1.591 = e^{4k}
\]

Take the natural logarithm of both sides:

\[
\ln(1.591) = 4k
\]

\[
k = \frac{\ln(1.591)}{4}
\]

Now calculate \( k \).

### Step 3: Plugging the values into the equation:

\[
k \approx \frac{\ln(1.591)}{4} \approx 0.1163
\]

### Step 4: Write the final equation:

Now that we have \( k \), the exponential function for \( W \) is:

\[
W(t) = 371.4 \cdot e^{0.1163t}
\]

This is the exponential model for worldwide wind generating capacity \( W \) as a function of time \( t \), where \( t \) represents the number of years since 2014.

---

Would you like more details on how to use this model, or have any questions about the process?

Here are 5 related questions for deeper exploration:
1. How can you use this model to predict wind capacity in 2025?
2. How do we calculate the doubling time for this exponential growth?
3. What happens if the growth rate changes over time?
4. How does this model compare to a linear growth model?
5. Can you express the capacity as a percentage increase per year?

**Tip**: The growth rate \( k \) is key in exponential functions; small changes in \( k \) can lead to significant changes over long periods!

The worldwide wind generating capacity W was 371.4 gigawatts at the end of 2014 and 591.1 gigawatts at the end of 2018 (t is years since 2014). Write W as an exponential function of t.

Learn how to model the worldwide wind generating capacity using an exponential function. The problem involves calculating the growth rate based on data from 2014 to 2018, utilizing key algebraic concepts and exponential functions. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation