Question

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the university of florida has a cattle research and education center in hardee county. in 2015, an economist at the center used data to create a quadratic the modeled past and projected florida feeder calf prices for the years 2015 - 2025.
the graph of the function, ( f(t)=1.6 t^{2}-19.6 t + 221.6 ), that models the data is shown on the coordinate grid.
determine the interval for which the price of a feeder calf is decreasing. write your answer using interval notation.

Explanation:

Step1: Identify the quadratic function form

The quadratic function is \(F(t)=1.6t^{2}-19.6t + 221.6\), where \(a = 1.6\), \(b=-19.6\), \(c = 221.6\). For a quadratic function \(y = ax^{2}+bx + c\), the axis of symmetry is given by \(t=-\frac{b}{2a}\).

Step2: Calculate the axis of symmetry

Substitute \(a = 1.6\) and \(b=-19.6\) into the formula \(t =-\frac{b}{2a}\).

$$ t=-\frac{-19.6}{2\times1.6}=\frac{19.6}{3.2}=6.125 $$

Since \(a = 1.6>0\), the parabola opens upwards. A quadratic function \(y = ax^{2}+bx + c\) (\(a>0\)) is decreasing on the interval \((-\infty,t)\) where \(t\) is the \(x\) - coordinate of the vertex.

Answer:

\((0,6.125)\)