Exercise 79 Page 64

Practice makes perfect

a We are given two quadratic functions that model the popping volume of popcorn.

Hot-air popping	Hot-oil popping
$f (x) = - 0.761 (x - 5.52) (x - 22.6)$	$g (x) = - 0.652 (x - 5.35) (x - 21.8)$

The moisture content that maximizes the popping volume corresponds to the $x -$ coordinate of the vertex, and the maximum volume corresponds to the $y -$ coordinate. Let's determine the vertex of $y = f (x) .$

Function	Vertex
$f (x) = a (x - p) (x - q)$	$x = \frac{p + q}{2}$	$y = f (\frac{p + q}{2})$
$f (x) = - 0.761 (x - 5.52) (x - 22.6)$	$x = \frac{5 . 5 2 + 2 2 . 6}{2} = 14.1$	$f (14.1) = 55.5$

The moisture content that maximizes the popping volume is about $14.1 %,$ and the maximum volume is about $55.5 cm^{3} / g .$

b As in Part A, the moisture content that maximizes the popping volume corresponds to the

x -

coordinate of the vertex, and the maximum volume corresponds to the

y -

coordinate. Let's determine the vertex of

y = g (x) .

Function	Vertex
$g (x) = a (x - p) (x - q)$	$x = \frac{p + q}{2}$	$y = f (\frac{p + q}{2})$
$g (x) = - 0.652 (x - 5.35) (x - 21.8)$	$x = \frac{5 . 3 5 + 2 1 . 8}{2} = 13.6$	$f (13.6) = 44.1$

Consequently, the moisture content that maximizes the popping volume is about $13.6 %,$ and the maximum volume is about $44.1 cm^{3} / g .$

c In this part we must graph both function using the graphing calculator.

Let's enter the equations into a graphing calculator by pushing $Y =$ and typing the right-hand side of the equations in the two first rows.

By pushing $GRAPH,$ the calculator will draw the equations.

For the functions to be visible on the screen, re-size the standard window by pushing the $WINDOW$ button. Change the settings to a more appropriate size and then push $GRAPH .$

Since the variables $x$ and $y$ represent units of measure they have to be greater than zero. Using this and the information obtained in the first two parts, we can state the following conclusions.

The domain for the hot-air popping $(f (x))$ is $5.52 \leq x \leq 22.6,$ and its range is $0 \leq y \leq 55.5 .$ It means that the moisture content for the kernels can be between $5.52 %$ and $22.6 %$ and the popping volume can be between $0$ to $55.5 cm^{3} / g .$
The domain for the hot-oil popping $(g (x))$ is $5.35 \leq x \leq 21.8,$ and its range is $0 \leq y \leq 44.1 .$ It means that the moisture content for the kernels can be between $5.35 %$ and $21.8 %$ and the popping volume can be between $0$ to $44.1 cm^{3} / g .$

Subchapter links

Communicate Your Answer p.55

Monitoring Progress p.57-60

Exercises p.61-64

Exercise 79 Page 64

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