Question

let ( f(t) ) be the fuel consumption in billions of gallons in ( t ) years since 1990. a quadratic model for the data is ( f(t)=-0.114 t^{2}+4.62 t + 127.598 ). use the above scatter plot to decide whether the quadratic model fits the data well. the function is not a good model for the data the function is a good model for the data. estimate the fuel consumption in the u. s. in 2013. billions of gallons. (please round to the nearest tenth) use the model to predict the (whole) year in which u.s. fuel consumption will peak. do not write decimals.

Explanation:

Step1: Find the value of \(t\) for 2013

Since \(t\) is years since 1990, for 2013, \(t = 2013 - 1990=23\)

Step2: Substitute \(t = 23\) into the function \(F(t)=- 0.114t^{2}+4.62t + 127.598\)

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Step3: Find the year when fuel consumption peaks

For a quadratic function \(y = ax^{2}+bx + c\) (\(a=-0.114\), \(b = 4.62\)), the \(t\)-value of the vertex is given by \(t=-\frac{b}{2a}\)

$$ t=-\frac{4.62}{2\times(- 0.114)}=\frac{4.62}{0.228}\approx20.26 $$

Since \(t\) represents years since 1990, the year is \(1990 + 20=2010\)

Answer:

173.6 billions of gallons; 2010