Question

the us crude oil production, in billions of barrels, for the years from 2005 projected to 2075, can be modeled by ( y=-0.001x^{2}+0.026x + 1.97 ), with ( x ) equal to the year after 2005 and ( y ) equal to the number of billions of barrels of crude oil.
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.
c. the minimum number of barrels of crude oil projected to be produced during this period is 14.500 billion barrels.
d. the maximum number of barrels of crude oil projected to be produced during this period is 14.500 billion barrels.
b. the model predicts the crude oil production to be 2.116 billion barrels of oil in 2028. (round to three decimal places as needed.)
c. choose the correct graph below

Explanation:

Step1: Find the vertex of the quadratic function

For a quadratic function \(y = ax^{2}+bx + c\), the \(x\) - coordinate of the vertex is given by \(x=-\frac{b}{2a}\).
Here, \(a=- 0.001\), \(b = 0.026\), \(c = 1.97\).
\(x=-\frac{0.026}{2\times(-0.001)}=\frac{-0.026}{-0.002}=13\)
Substitute \(x = 13\) into the function \(y=-0.001x^{2}+0.026x + 1.97\)
\(y=-0.001\times(13)^{2}+0.026\times13 + 1.97\)
\(y=-0.001\times169+0.338+1.97\)
\(y=-0.169+0.338 + 1.97\)
\(y=2.139\)

Step2: Interpret the vertex

Since \(a=-0.001<0\), the parabola opens downwards. The vertex \((13,2.139)\) means that in the year \(1995 + 13=2008\), the maximum number of billions of barrels of crude oil is \(2.139\) billion barrels.

Step3: Predict the production in 2028

If \(x\) is the year after 1995, for the year 2028, \(x=2028 - 1995=33\)
Substitute \(x = 33\) into the function \(y=-0.001x^{2}+0.026x + 1.97\)
\(y=-0.001\times(33)^{2}+0.026\times33+1.97\)
\(y=-0.001\times1089+0.858+1.97\)
\(y=-1.089+0.858+1.97\)
\(y=1.739\) (rounded to three decimal places)

Answer:

a. The vertex is \((13, 2.139)\). It means that in 2008, the maximum number of billions of barrels of crude oil is \(2.139\) billion barrels.
b. \(1.739\)
c. (Since the function is a parabola opening downwards with vertex at \(x = 13\), we need to plot points for \(x\) values from \(0\) (2005) to \(30\) (2025). For \(x = 0\), \(y=-0.001\times0^{2}+0.026\times0+1.97 = 1.97\); for \(x=10\), \(y=-0.001\times100+0.026\times10 + 1.97=- 0.1+0.26+1.97=2.13\); for \(x = 20\), \(y=-0.001\times400+0.026\times20+1.97=-0.4 + 0.52+1.97=2.09\); for \(x=30\), \(y=-0.001\times900+0.026\times30+1.97=-0.9+0.78+1.97=1.85\). The correct graph is the one that is a parabola opening downwards passing through these points approximately)