The U.S. Crude Oil​ production, in billions of​ barrels, for the years from 2005 projected to 2025​, can be modeled y equals negative 0.002 x squared plus 0.043 x plus 1.949​, with x equal to the years after 2005 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 2028​?
c. Graph the function for the years 2005 to 2025.
Question content area bottom
Part 1
a. The vertex of the graph of this model is vequals​(
  
10.750​,
  
2.180​). ​(Round to three decimal places as​ needed.)
Part 2
Choose the correct interpretation of the vertex below.
A.
The maximum number of barrels of crude oil projected to be produced during this period is 10.750 billion barrels.
B.
The maximum number of barrels of crude oil projected to be produced during this period is 2.180 billion barrels.
C.
The minimum number of barrels of crude oil projected to be produced during this period is 10.750 billion barrels.
D.
The minimum number of barrels of crude oil projected to be produced during this period is 2.180 billion barrels.

Answer 1

To solve the problem step by step:

Part 1: Finding and Interpreting the Vertex

The vertex of a quadratic function in the form of \(y = ax^2 + bx + c\) can be found using the vertex formula: \[ x = -\frac{b}{2a} \]

Given the model: \[ y = -0.002x^2 + 0.043x + 1.949 \] Here, \(a = -0.002\) and \(b = 0.043\).

Calculating the x-coordinate of the vertex: \[ x = -\frac{0.043}{2 \times -0.002} = -\frac{0.043}{-0.004} = 10.75 \]

Now, we find the y-coordinate by substituting \(x = 10.75\) into the equation: \[ y = -0.002(10.75)^2 + 0.043(10.75) + 1.949 \] \[ = -0.002(115.5625) + 0.046225 + 1.949 \] \[ = -0.231125 + 0.046225 + 1.949 \] \[ = 1.764875 \approx 1.765 \text{ (rounded to three decimal places)} \]

Thus, the vertex is approximately: \[ v \approx (10.750, 1.765) \]

The interpretation of the vertex is as follows:

• The x-coordinate \(10.750\) represents years after 2005, corresponding to the year \(2005 + 10.75 = 2016.75\) (around mid-2016).
• The y-coordinate \(1.765\) (in billions of barrels) represents the maximum projected production of crude oil.

Interpretation Choice

The correct interpretation of the vertex is:

• A. The maximum number of barrels of crude oil projected to be produced during this period is 1.765 billion barrels.

Part 2: Crude Oil Production in 2028

To find the predicted production for 2028, we calculate \(x\) for the year \(2028\): \[ x = 2028 - 2005 = 23 \]

Substituting \(x = 23\) into the model: \[ y = -0.002(23)^2 + 0.043(23) + 1.949 \] \[ = -0.002(529) + 0.989 + 1.949 \] \[ = -1.058 + 0.989 + 1.949 \] \[ = 1.880 \text{ billion barrels (approximately)} \]

Part 3: Graphing the Function for 2005 to 2025

To graph the function \(y = -0.002x^2 + 0.043x + 1.949\) for \(x\) from \(0\) (2005) to \(20\) (2025):

1. Plot the vertex, which is at approximately \((10.75, 1.765)\).
2. Evaluate the function at several points in the interval (e.g., \(0, 5, 10, 15, 20\)) to get corresponding \(y\) values:
   • \(x = 0\) gives \(y = 1.949\)
   • \(x = 5\) gives \(y \approx 1.881\)
   • \(x = 10\) gives \(y \approx 1.766\)
   • \(x = 15\) gives \(y \approx 1.601\)
   • \(x = 20\) gives \(y \approx 1.287\)
3. The graph will show a downward-opening parabola with a peak at the vertex.

In conclusion, the answers to all parts are:

Part 1: Vertex is \((10.750, 1.765)\) with interpretation choice A.

Part 2: Predicted production for 2028 is approximately \(1.880\) billion barrels.

Part 3: Graph is a downward-opening parabola between the years 2005 and 2025.