a couple of college students get the brilliant idea to open a refreshment stand at the regatta festival this…

a couple of college students get the brilliant idea to open a refreshment stand at the regatta festival this summer. the students sell freshly squeezed lemonade. they pay the city$572 to set up their stand for the entire regatta. they also figure the cost for lemons, ice, and other things they need is $0.40 per glass. they sell each glass for $1.50. define units for the cost to make the lemonade and their gross income. enter a variable for the number of glasses sold and use this variable to write expressions for the cost to make the lemonade and their gross income. 1. what is the initial cost? what is the initial income? 2. what is the cost and the income obtained from selling one hundred ten glasses? 3. how many glasses must they sell to break even, according to the algebraic model? 4. if the income from selling the lemonade was ninety dollars, how many glasses did they sell? after completing the worksheet, graph your model.

a couple of college students get the brilliant idea to open a refreshment stand at the regatta festival this summer. the students sell freshly squeezed lemonade. they pay the city$572 to set up their stand for the entire regatta. they also figure the cost for lemons, ice, and other things they need is $0.40 per glass. they sell each glass for $1.50. define units for the cost to make the lemonade and their gross income. enter a variable for the number of glasses sold and use this variable to write expressions for the cost to make the lemonade and their gross income. 1. what is the initial cost? what is the initial income? 2. what is the cost and the income obtained from selling one hundred ten glasses? 3. how many glasses must they sell to break even, according to the algebraic model? 4. if the income from selling the lemonade was ninety dollars, how many glasses did they sell? after completing the worksheet, graph your model.

Answer

Explanation:

Step1: Define cost and income functions

Let $x$ be the number of glasses sold. The fixed - cost of setting up the stand is $572$ and the variable cost per glass is $0.40$. So the cost function $C(x)=572 + 0.4x$. The selling price per glass is $1.50$, so the income function $I(x)=1.5x$.

Step2: Calculate cost and income for 110 glasses

For the cost when $x = 110$, substitute $x$ into the cost function: $C(110)=572+0.4\times110=572 + 44=616$. For the income when $x = 110$, substitute $x$ into the income function: $I(110)=1.5\times110 = 165$.

Step3: Find the break - even point

The break - even point occurs when $C(x)=I(x)$. So we set $572+0.4x=1.5x$. Subtract $0.4x$ from both sides: $572=1.5x - 0.4x$, which simplifies to $572 = 1.1x$. Then solve for $x$: $x=\frac{572}{1.1}=520$.

Step4: Find the number of glasses sold for an income of 90 dollars

Set $I(x)=90$. Since $I(x)=1.5x$, we have $1.5x=90$. Solve for $x$ by dividing both sides by $1.5$: $x=\frac{90}{1.5}=60$.

Answer:

  1. Initial cost: $572$, Initial income: $0$
  2. Cost: $616$, Income: $165$
  3. $520$
  4. $60$