Question

watch the video and then solve the problem given below.
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the u.s. crude oil production, in billions of barrels, for the years from 2015 projected to 2035, can be modeled ( y=-0.002 x^{2}+0.057 x + 1.577 ), with ( x ) equal to the years after 2015 and ( y ) equal to the number of billions of barrels of crude oil. answer parts a through c.

find and interpret the vertex of the graph of this model.
the vertex of the graph of this model is ( v= ) (type integers or decimals rounded to the nearest thousandth as needed)

Explanation:

Step1: Identify the coefficients

For the quadratic function \(y = ax^{2}+bx + c\), in the given model \(y=-0.002x^{2}+0.057x + 1.577\), we have \(a=-0.002\), \(b = 0.057\), and \(c = 1.577\).

Step2: Calculate the \(x\) - coordinate of the 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 \(a=-0.002\) and \(b = 0.057\) into the formula:

$$x=-\frac{0.057}{2\times(-0.002)}=\frac{0.057}{0.004}=14.25$$

Step3: Calculate the \(y\) - coordinate of the vertex

Substitute \(x = 14.25\) into the function \(y=-0.002x^{2}+0.057x + 1.577\).

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Answer:

The vertex of the graph of this model is \(v=(14.25,1.983)\). This means that in the year \(2015 + 14.25=2029.25\) (approximately 2029), the U.S. crude oil production is at a maximum (since \(a=-0.002<0\), the parabola opens downwards) of approximately \(1.983\) billion barrels.