Question

the u s. crude oil production, in billions of barrels, for the years from 2005 projected to 2025, can be modeled ( y = - 0.001 x ^ { 2 } + 0.029 x + 1.978 ), with ( x ) equal to the years 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.
a. the vertex of the graph of this model is ( v = (square,square) ) (round to three decimal places as needed)

Explanation:

Step1: Identify coefficients

For the quadratic function \(y = ax^{2}+bx + c\), here \(a=-0.001\), \(b = 0.029\), \(c = 1.978\).

Step2: Find \(x\) - coordinate of vertex

The formula for the \(x\) - coordinate of the vertex of a quadratic function \(y=ax^{2}+bx + c\) is \(x=-\frac{b}{2a}\).
Substitute \(a=-0.001\) and \(b = 0.029\) into the formula:
\(x=-\frac{0.029}{2\times(- 0.001)}=\frac{0.029}{0.002}=14.5\)

Step3: Find \(y\) - coordinate of vertex

Substitute \(x = 14.5\) into the function \(y=-0.001x^{2}+0.029x + 1.978\)
\(y=-0.001\times(14.5)^{2}+0.029\times14.5+1.978\)
First, calculate \((14.5)^{2}=210.25\)
\(y=-0.001\times210.25+0.029\times14.5 + 1.978\)
\(y=-0.21025+0.4205+1.978\)
\(y=2.18825\approx2.188\)

Answer:

The vertex of the graph of this model is \(v=(14.5,2.188)\). This means that 14.5 years after 2005 (i.e., in the year \(2005 + 14.5=2019.5\)), the U.S. crude oil production is at a maximum (since \(a=-0.001<0\), the parabola opens downwards) of approximately \(2.188\) billion barrels.