Question

the function $f(x) = \frac{30 + 0.25x}{x}$ yields the average cost in dollars per cup of lemonade made by lincoln at his lemonade stand when he makes $x$ cups of lemonade. which statement best fits the situation modeled by the function? lincoln sells lemonade for 25 cents a cup. lincoln uses 30 cents worth of supplies per cup. lincoln gave away 30 cups of lemonade for free. lincoln uses 25 cents worth of supplies per cup.

Explanation:

Step1: Analyze the function components

The function is \( f(x)=\frac{30 + 0.25x}{x} \), which can be split into \( \frac{30}{x}+\frac{0.25x}{x}=\frac{30}{x}+ 0.25 \). Here, \( 0.25x \) represents the total cost for the variable supplies (per cup cost times number of cups), so per cup, the variable supply cost is \( 0.25 \) dollars (or 25 cents), and \( 30 \) is a fixed cost (like initial setup or equipment cost) that gets divided by the number of cups \( x \) to find its contribution to the average cost per cup.

Step2: Evaluate each statement

• "Lincoln sells lemonade for 25 cents a cup": The function is about cost, not selling price, so this is incorrect.
• "Lincoln uses 30 cents worth of supplies per cup": The fixed cost is 30 dollars (not cents) and is divided by \( x \), so this is incorrect.
• "Lincoln gave away 30 cups of lemonade for free": The function has no relation to giving away cups, so this is incorrect.
• "Lincoln uses 25 cents worth of supplies per cup": The \( 0.25x \) term, when divided by \( x \), gives 0.25 dollars (25 cents) per cup for variable supplies, so this matches the per - cup variable cost from the function.

Answer:

Lincoln uses 25 cents worth of supplies per cup.