Using data from 2010 and projected to 2030, the U.S. crude oil production, in billions of barrels, can be approximated by

*P* = −0.00105*t*2 + 0.0367*t* + 1.94

Estimate the production, in billions of barrels, for 2016.

_____ billion barrels

To estimate the U.S. crude oil production in 2016 using the given quadratic equation:

\[
P(t) = -0.00105t^2 + 0.0367t + 1.94
\]

where \( t \) is the number of years since 2010, we first determine the value of \( t \) for the year 2016:

\[
t = 2016 - 2010 = 6
\]

Now, substitute \( t = 6 \) into the equation to find \( P(6) \):

\[
P(6) = -0.00105(6)^2 + 0.0367(6) + 1.94
\]

\[
P(6) = -0.00105(36) + 0.0367(6) + 1.94
\]

\[
P(6) = -0.0378 + 0.2202 + 1.94
\]

\[
P(6) = 2.1224
\]

So, the estimated U.S. crude oil production for 2016 is approximately **2.12 billion barrels**.

Would you like further details or have any questions?

Here are five related questions to explore further:
1. How would the estimated production change if the coefficient of \( t^2 \) were different?
2. What is the significance of the constant term in the quadratic equation?
3. Can this model be used reliably to predict production for years beyond 2030?
4. How would the equation change if we had data extending beyond 2030?
5. What would the production be estimated for the year 2020?

**Tip:** When dealing with quadratic equations, the sign of the coefficient of \( t^2 \) determines whether the parabola opens upwards or downwards, which affects the prediction behavior of the model over time.

Learn how to estimate U.S. crude oil production using a quadratic equation based on data from 2010 projected to 2030. Explore the step-by-step solution to find the estimated production for the year 2016, approximately 2.12 billion barrels.

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