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

where *t* is the number of years after 2010.†

(a)

What *t*-value represents 2016?

*t* =  

(b)

Actual production in 2014 was 2.10 billion barrels. What does the equation give as the 2014 approximation in billions of barrels?

_______  billion barrels

(c)

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

 ______mbillion barrels

Let's break down the problem and solve it step by step.

### Given:
The equation for the U.S. crude oil production is:
\[
P(t) = -0.00105t^2 + 0.0367t + 1.94
\]
where \( t \) is the number of years after 2010.

#### (a) What \( t \)-value represents 2016?

To calculate \( t \), we use the formula \( t = \text{year} - 2010 \).

For 2016:
\[
t = 2016 - 2010 = 6
\]

So, the \( t \)-value representing 2016 is \( \mathbf{6} \).

#### (b) What does the equation give as the 2014 approximation in billions of barrels?

For 2014, \( t = 2014 - 2010 = 4 \).

We substitute \( t = 4 \) into the equation:
\[
P(4) = -0.00105(4)^2 + 0.0367(4) + 1.94
\]
Let's calculate this value.

The equation gives an approximation of \( \mathbf{2.07} \) billion barrels for 2014.

#### (c) Estimate the production, in billions of barrels, for 2016.

For 2016, \( t = 6 \). We substitute \( t = 6 \) into the equation:
\[
P(6) = -0.00105(6)^2 + 0.0367(6) + 1.94
\]
Let's calculate this value.The estimated crude oil production for 2016, according to the equation, is approximately \( \mathbf{2.12} \) billion barrels.

### Summary:
- (a) The \( t \)-value representing 2016 is \( 6 \).
- (b) The equation gives an approximation of \( 2.07 \) billion barrels for 2014.
- (c) The estimated production for 2016 is \( 2.12 \) billion barrels.

Do you want further details or have any questions?

Here are 5 related questions for further exploration:
1. How does the quadratic nature of the equation affect the production over time?
2. What would be the projected production in 2020 according to the model?
3. How close is the model’s estimate for 2014 compared to the actual production?
4. What does the negative coefficient for the quadratic term imply about long-term production trends?
5. How would changes in the coefficients impact future predictions?

**Tip:** Quadratic models are often used to represent trends that peak and then decline over time.

Explore the approximation of U.S. crude oil production from 2010 to 2030 using a quadratic equation model. Includes calculations for 2016 and 2014 production estimates based on the model equation.

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