The U.S. Crude Oil​ production, in billions of​ barrels, for the years from 2015 projected to 2035​, can be modeled y equals negative 0.003 x squared plus 0.053 x plus 1.992​, with x equal to the years after 2015 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 2038​?
c. Graph the function for the years 2015 to 2035.
Question content area bottom
Part 1
a. The vertex of the graph of this model is vequals​(
  
enter your response here​,
  
enter your response here​). ​(Round to three decimal places as​ needed.)

Answer 1

To find the vertex of the quadratic function given by the equation:

\[ y = -0.003x^2 + 0.053x + 1.992 \]

we can use the formula for the x-coordinate of the vertex of a parabola represented by the equation \( y = ax^2 + bx + c \):

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

Here, \( a = -0.003 \) and \( b = 0.053 \).

Step 1: Calculate the x-coordinate of the vertex

\[ x = -\frac{0.053}{2 \cdot -0.003} = -\frac{0.053}{-0.006} \approx 8.8333 \]

Step 2: Calculate the y-coordinate of the vertex

Now substitute this x value back into the original equation to find the y-coordinate:

\[ y = -0.003(8.8333)^2 + 0.053(8.8333) + 1.992 \]

Calculating \( (8.8333)^2 \):

\[ 8.8333^2 \approx 78.0735 \]

Now substituting into \( y \):

\[ y \approx -0.003 \cdot 78.0735 + 0.053 \cdot 8.8333 + 1.992 \]

\[ y \approx -0.2342 + 0.4687 + 1.992 \]

\[ y \approx 1.2265 \]

Vertex Interpretation

Thus, the vertex is approximately:

\[ \text{Vertex} = (8.833, 1.227) \]

Answer for Part a:

The vertex of the graph of this model is \( v = (8.833, 1.227) \).

Interpretation: The vertex indicates that at approximately 8.833 years after 2015 (which is around the year 2024), the crude oil production is projected to reach around 1.227 billion barrels.

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Part b: Predicting crude oil production in 2038

To find the production in 2038, we calculate \( x \) for that year. Since \( x \) represents the years after 2015:

\[ x = 2038 - 2015 = 23 \]

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

\[ y = -0.003(23)^2 + 0.053(23) + 1.992 \]

Calculating \( (23)^2 \):

\[ 23^2 = 529 \]

Now substituting into \( y \):

\[ y = -0.003 \cdot 529 + 0.053 \cdot 23 + 1.992 \]

\[ y \approx -1.587 + 1.219 + 1.992 \]

\[ y \approx 1.624 \]

Prediction for Part b:

The model predicts that the crude oil production in 2038 will be approximately 1.624 billion barrels.

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Part c: Graphing the Function

To graph the function \( y = -0.003x^2 + 0.053x + 1.992 \) for the years from 2015 to 2035, we consider \( x \) values from 0 (2015) to 20 (2035). Use points such as:

• Calculate \( y \) for integers \( x = 0, 1, 2, \ldots, 20 \).
• You can create a table of \( x \) values (years after 2015) and their corresponding \( y \) (billions of barrels).
• Finally, plot these points on a graph with \( x \) on the horizontal axis and \( y \) on the vertical axis.

Note: You could use software, a graphing calculator, or a spreadsheet application to plot this for better accuracy and visualize the function.

If you need assistance with graphing using specific tools or software, feel free to ask!