The function $f(x)=\frac{30+0.25 x}{x}$ 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?
A. Lincoln sells lemonade for 25 cents a cup.
B. Lincoln gave away 30 cups of lemonade for free.
C. Lincoln uses 25 cents worth of supplies per cup.
D. Lincoln uses 30 cents worth of supplies per cup.
Answer (2)
Rewrite the function as f ( x ) = x 30 + 0.25 .
Interpret x 30 as the fixed cost divided by the number of cups.
Interpret 0.25 as the variable cost per cup.
The statement that best fits the situation is: Lincoln uses 25 cents worth of supplies per cup. L in co l n u ses 25 ce n t s w or t h o f s u ppl i es p erc u p .
Explanation
Understanding the Function We are given the function f ( x ) = f r a c 30 + 0.25 x x , which represents the average cost per cup of lemonade when Lincoln makes x cups. We need to interpret this function in the context of the given options.
Rewriting the Function Let's rewrite the function to better understand its components: f ( x ) = f r a c 30 x + f r a c 0.25 x x = f r a c 30 x + 0.25
Interpreting the Terms The function f ( x ) now consists of two terms: f r a c 30 x and 0.25 . The term f r a c 30 x represents the fixed cost of 30 dollars divided by the number of cups x . This suggests a fixed initial cost. The term 0.25 represents a cost of 0.25 dollars per cup, which is a variable cost.
Analyzing the Options Now, let's analyze the given options:
Lincoln sells lemonade for 25 cents a cup. This option relates to revenue, not cost, so it's not the correct interpretation of the function.
Lincoln gave away 30 cups of lemonade for free. This doesn't directly relate to the structure of the cost function.
Lincoln uses 25 cents worth of supplies per cup. This matches the 0.25 term in the function, representing the variable cost per cup.
Lincoln uses 30 cents worth of supplies per cup. This doesn't match the 0.25 term in the function.
Connecting to the Options The term 0.25 in the function f ( x ) = f r a c 30 x + 0.25 represents the cost of supplies per cup, which is 25 cents. The term f r a c 30 x represents a fixed cost of 30 dollars, which could be for initial setup or equipment.
Final Answer Based on the analysis, the statement that best fits the situation modeled by the function is: Lincoln uses 25 cents worth of supplies per cup.
Examples
Understanding cost functions like this can help businesses make informed decisions about pricing and production. For example, if Lincoln knows his fixed costs are $30 and his variable costs are 0.25 p erc u p , h ec an u se t hi s f u n c t i o n t o d e t er min e h o w man yc u p so f l e m o na d e h e n ee d s t ose llt o b re ak e v e n or t ore a c ha cer t ain p ro f i tt a r g e t . T hi s t y p eo f ana l ys i s i scr u c ia l f or an y b u s in ess t o mana g e i t s f inan cese ff ec t i v e l y . B y ana l yz in g t h ecos t f u n c t i o n f(x) = \frac{30}{x} + 0.25$, Lincoln can optimize his lemonade stand operations, ensuring profitability and sustainability.
Lincoln's function for average cost per cup of lemonade shows that he incurs a variable cost of 25 cents per cup. The best statement reflecting this setup is that Lincoln uses 25 cents worth of supplies per cup. Therefore, the correct answer is option C.
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