World consumption of a certain metal is running at the rate of
337e^0.02t
thousand metric tons per year, where t is measured in years and
t = 0
corresponds to 2014. Find the total consumption of the metal from 2014 to 2029. (Round your answer to the nearest whole number.)

To find the total consumption of the metal from 2014 to 2029, we need to calculate the definite integral of the consumption rate function over this period. The given consumption rate is \( 337e^{0.02t} \) thousand metric tons per year, where \( t = 0 \) corresponds to 2014.

The total consumption \( C \) over the period from \( t = 0 \) to \( t = 15 \) (since 2029 - 2014 = 15) is given by:

\[
C = \int_{0}^{15} 337e^{0.02t} \, dt
\]

Let's calculate this integral:

\[
C = 337 \int_{0}^{15} e^{0.02t} \, dt
\]

To integrate \( e^{0.02t} \), we use the formula \( \int e^{kt} \, dt = \frac{1}{k}e^{kt} + C \). Thus, 

\[
\int e^{0.02t} \, dt = \frac{1}{0.02} e^{0.02t} = 50 e^{0.02t}
\]

Therefore,

\[
C = 337 \times 50 \left[ e^{0.02t} \right]_{0}^{15}
\]

\[
C = 16850 \left( e^{0.02 \times 15} - e^{0.02 \times 0} \right)
\]

\[
C = 16850 \left( e^{0.3} - 1 \right)
\]

Now, we compute the value of \( e^{0.3} \) and the total consumption \( C \).The total consumption of the metal from 2014 to 2029 is approximately **5895 thousand metric tons**. This value is rounded to the nearest whole number.

Would you like more details or have any questions? 

Here are some follow-up questions:

1. What is the formula for the derivative of the consumption rate?
2. How does the consumption rate change over time?
3. What would be the total consumption if the rate increased by 0.01 instead of 0.02?
4. How would the total consumption change if the starting year was different?
5. What is the average consumption rate over the period?
6. How do you calculate the definite integral of an exponential function?
7. What is the significance of the parameter 0.02 in the consumption rate formula?
8. How would you find the total consumption over a different time period?

**Tip:** When calculating integrals, always ensure you understand the function's behavior over the interval to check the plausibility of the result.

Learn how to calculate the total consumption of a metal from 2014 to 2029 using the definite integral of an exponential consumption rate function. Understand the formula and steps involved in finding the total consumption, rounded to the nearest whole number.

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