The U.S. Crude Oil​ production, in billions of​ barrels, for the years from 2015 projected to 2035​, can be modeled y equals negative 0.002 x squared plus 0.033 x plus 1.957​, 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.
he vertex of the graph of this model is vequals​(
  
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Answer 1

To solve the given problem, we will analyze the quadratic function modeled by \( y = -0.002x^2 + 0.033x + 1.957 \).

Part a: Find and interpret the vertex

The \( x \)-coordinate of the vertex of a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula:

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

Here, \( a = -0.002 \) and \( b = 0.033 \).

1. Calculate the \( x \)-coordinate of the vertex:

\[ x = -\frac{0.033}{2 \times -0.002} = -\frac{0.033}{-0.004} = 8.25 \]

2. Substitute \( x \) back into the equation to find the \( y \)-coordinate of the vertex:

\[ y = -0.002(8.25)^2 + 0.033(8.25) + 1.957 \] \[ y = -0.002(68.0625) + 0.27225 + 1.957 \] \[ y = -0.136125 + 0.27225 + 1.957 \] \[ y \approx 2.093875 \approx 2.094 \text{ (rounded to three decimal places)} \]

Thus, the vertex of the graph is:

\[ \text{Vertex} = (8.25, 2.094) \]

Interpretation: The vertex \((8.25, 2.094)\) indicates that, approximately 8.25 years after 2015 (which is around the year 2023), the crude oil production is at its maximum of about 2.094 billion barrels.

Part b: Prediction for crude oil production in 2038

To find the production in 2038, we calculate \( x \) when the year is 2038. Since \( x = \text{year} - 2015 \):

\[ x = 2038 - 2015 = 23 \]

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

\[ y = -0.002(23)^2 + 0.033(23) + 1.957 \] \[ y = -0.002(529) + 0.759 + 1.957 \] \[ y = -1.058 + 0.759 + 1.957 \] \[ y \approx 1.658 \]

So, the model predicts the crude oil production in 2038 will be approximately 1.658 billion barrels.

Part c: Graph the function for the years 2015 to 2035

To graph the function from 2015 to 2035, you need to plot the function for \( x \) ranging from 0 to 20 (since \( x = 0 \) corresponds to 2015 and \( x = 20 \) corresponds to 2035).

You can generate specific \( y \) values for \( x = 0, 1, 2, ... , 20 \) using the function to create points for the graph.

Here are the key points you could compute with the equation for better insight:

• \( y = 1.957 \) when \( x = 0 \) (2015)
• \( y = 1.968 \) when \( x = 1 \) (2016)
• ...
• The maximum point approximately at \( (8.25, 2.094) \) around 2023
• Then values decrease again towards \( y = 1.145 \) when \( x = 20 \) (2035)

You may use graphing software or a calculator for more accuracy and clarity in rendering the graph visually from 2015 to 2035.

If you need specific point values calculated or further instructions for graphing, feel free to ask!