The U.S. Crude Oil​ production, in billions of​ barrels, for the years from 2010 projected to 2030​, can be modeled y equals negative 0.003 x squared plus 0.055 x plus 1.996​, with x equal to the years after 2010 and y equal to the number of billions of barrels of crude oil.

Question

a. Find and interpret the vertex of the graph of this model.
b. What does the model predict the crude oil production will be in 2033​?
c. Graph the function for the years 2010 to 2030.

Answer 1

To solve these parts, we will work through the quadratic function given:

\[ y = -0.003x^2 + 0.055x + 1.996 \]

where \( x \) is the number of years after 2010, and \( y \) is the crude oil production in billions of barrels.

Part a: Finding the Vertex

The vertex of a parabola defined by a quadratic function \( y = ax^2 + bx + c \) can be found using the formula:

\[ x_v = -\frac{b}{2a} \]

In our case:

• \( a = -0.003 \)
• \( b = 0.055 \)

Calculating \( x_v \):

\[ x_v = -\frac{0.055}{2 \times -0.003} = -\frac{0.055}{-0.006} \approx 9.1667 \]

Now, to find \( y_v \), we substitute \( x_v \) back into the equation:

\[ y_v = -0.003(9.1667)^2 + 0.055(9.1667) + 1.996 \]

Calculating \( (9.1667)^2 \):

\[ (9.1667)^2 \approx 84.0507 \]

Now, substituting back:

\[ y_v \approx -0.003(84.0507) + 0.055(9.1667) + 1.996 \] \[ y_v \approx -0.2522 + 0.5042 + 1.996 \approx 2.248 \]

Thus, the vertex is approximately \( (9.1667, 2.248) \).

Interpretation: The x-value of approximately 9.1667 corresponds to the year 2019 (2010 + 9.1667 ≈ 2019), and the maximum crude oil production \( y_v \) is approximately 2.248 billion barrels. This means that the model predicts that crude oil production will reach its peak in 2019 at about 2.248 billion barrels.

Part b: Predicting the crude oil production in 2033

To find the model's prediction for crude oil production in 2033, we first determine \( x \) for 2033:

\[ x = 2033 - 2010 = 23 \]

Now we substitute \( x = 23 \) into the equation:

\[ y = -0.003(23)^2 + 0.055(23) + 1.996 \] Calculating \( (23)^2 \):

\[ (23)^2 = 529 \]

Substituting back:

\[ y = -0.003(529) + 0.055(23) + 1.996 \] \[ y = -1.587 + 1.265 + 1.996 \approx 1.674 \]

So the model predicts that crude oil production in 2033 will be approximately 1.674 billion barrels.

Part c: Graphing the Function from 2010 to 2030

To graph the function \( y = -0.003x^2 + 0.055x + 1.996 \) for \( x \) from 0 (2010) to 20 (2030), we can calculate \( y \) values at several key points.

• At \( x = 0 \): \[ y = 1.996 \]
• At \( x = 5 \): \[ y = -0.003(5^2) + 0.055(5) + 1.996 = -0.075 + 0.275 + 1.996 \approx 2.196 \]
• At \( x = 10 \): \[ y \approx 2.248 \quad (\text{We already calculated this for the vertex.}) \]
• At \( x = 15 \): \[ y = -0.003(15^2) + 0.055(15) + 1.996 = -0.675 + 0.825 + 1.996 \approx 2.146 \]
• At \( x = 20 \): \[ y = -0.003(20^2) + 0.055(20) + 1.996 = -1.2 + 1.1 + 1.996 \approx 1.896 \]

Plotting these points on a graph or using a graphing utility will show a downward-opening parabola since \( a < 0 \).

You would find the crude oil production peaks around 2019 (at \( x \approx 9.1667 \)) and then it starts declining towards 2030.

In conclusion, the model suggests the production will peak in 2019 and gradually decline through 2030 onwards, hitting approximately 1.896 billion barrels.