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.

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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!