Exercise 11 Page 224

We will begin by defining the variables for the situation. The total profit can be found by multiplying the price per video by the number of videos. We will organize the given information on a table and write the new price and new number of videos depending on $x.$

Verbal Expression	Algebraic Expression
Increasing the price $x$ times ($\$$)	$0.25x$
New price	$2.25+0.25x$
Decreasing the number of videos $x$ times	$100x$
New number of videos	$1400-100x$
Total profit is $\$y.$	$y=(2.25+0.25x)(1400-100x)$

Now, we can write the equation in the form of $y=a{x}^2+bx+c$ to determine its characteristics. The maximum value will be at the vertex. The $x\text{-}$coordinate of the vertex tells us how many times we should increase the price to maximize the profit. We will use the formula for the axis of symmetry, $x=\text{-}\frac{b}{2a},$ to find the $x\text{-}$coordinate of the vertex. In this case, $a=\text{-}25$ and $b=125.$ The $x$ value tells us that if the store increases the price $2.5$ times, the income will be maximum. However, the number of increases must be a whole number. Therefore, the store should increase the price either $2$ times or $3$ times. We defined the new price on the table. By substituting $2$ and $3$ in the expression, we can find the possible prices.

Number of Decreases	$2.25+0.25x$	Price
$2$	$2.25+0.25(2)$	$2.75$
$3$	$2.25+0.25(3)$	$3$

Therefore, both $\$2.75$ and $\$3$ maximize the income.