Question

select the correct option from each drop - down menu.
sully manages a cafe that typically sells 240 cups of coffee per day for $2 each. sully knows that for each $0.25 increase in the price of coffee, 20 fewer cups of coffee will be sold. let x represent the number of $0.25 price increases and f(x) represent the total revenue earned from coffee sales.
sully wants to write an equation in the form that reveals the best potential earnings for the cafe in terms of revenue earned from coffee sales each day.
complete the following statement for this situation.
the function
reveals that the
revenue earned by the cafe from coffee sales will be $
after
$0.25 price increases.
reset
next

Explanation:

Step 1: Define Variables and Price - Quantity Relationship

Let \( x \) be the number of \(\$0.25\) price increases. The original price per cup is \(\$2\), so the new price per cup after \( x \) increases is \( p(x)=2 + 0.25x \). The original number of cups sold per day is 240, and for each \(\$0.25\) increase, 20 fewer cups are sold, so the number of cups sold after \( x \) increases is \( q(x)=240-20x \).

Step 2: Revenue Function

Revenue \( R(x) \) (which is \( f(x) \) in the problem) is the product of price per cup and number of cups sold. So, \( f(x)=p(x)\times q(x)=(2 + 0.25x)(240 - 20x) \).

Let's expand this function:
\[

$$\begin{align*} f(x)&=2\times240-2\times20x+0.25x\times240-0.25x\times20x\\ &=480-40x + 60x-5x^{2}\\ &=- 5x^{2}+20x + 480 \end{align*}$$

\]

If we want to find the revenue after, say, a certain number of price increases (the problem seems to have a drop - down for \( x \), but since the exact value of \( x \) from the drop - down is not fully visible, let's assume we want to find the revenue function or maybe for a specific \( x \). But generally, the function that reveals the revenue is \( f(x)=(2 + 0.25x)(240 - 20x) \) or the expanded form \( f(x)=-5x^{2}+20x + 480 \).

If we assume \( x = 1 \) (one \(\$0.25\) increase), then the price is \( 2+0.25\times1 = 2.25 \) dollars, the number of cups sold is \( 240 - 20\times1=220 \) cups, and the revenue is \( 2.25\times220 = 495 \) dollars. If \( x = 2 \), price is \( 2 + 0.5=2.5 \) dollars, number of cups is \( 240-40 = 200 \) cups, revenue is \( 2.5\times200 = 500 \) dollars.

But since the problem is about writing the function that reveals the revenue, the function is \( f(x)=(2 + 0.25x)(240 - 20x) \) (or the expanded quadratic function).

Answer:

The function that reveals the revenue earned by the cafe from coffee sales is \( \boldsymbol{f(x)=(2 + 0.25x)(240 - 20x)} \) (or \( \boldsymbol{f(x)=-5x^{2}+20x + 480} \) after expansion). If we need to find the revenue for a specific number of price increases (say \( x = n \)), we substitute \( n \) into the function. For example, if \( x = 0 \) (no price increase), \( f(0)=2\times240 = 480 \) dollars. If \( x = 2 \), \( f(2)=(2 + 0.5)(240 - 40)=2.5\times200 = 500 \) dollars.