Question

In the summertime your friend wants to have a snow cone stand at a local grocery store parking lot. You want to make sure he actually makes money. The grocery store charges $300 per month. All products are calculated to be 25 cents per customer. Your plans are to sell each cone for $1.25.
a) Write a mathematical expression representing the cost of operation for a month.

b) Write a mathematical expression representing sales.

c) Create an equation that will represent overall profit of the operation for a month

d) Simplify the equation into slope intercept form and standard form. What is the slope and explain its meaning in the context of selling snow cones? What is the intercept and what is its meaning?

e) Describe the method of graphing the slope intercept form and also the standard form of the equation.

f) What is the domain and range of the equation? Is the relation a function, why?

g) How many snow cones do you need to sell to break even in the first month?

h) How much is the profit if you sell 425, 550 or 700 snow cones in the first month of business?

Answer 1

a) The total monthly cost is the sum of the fixed cost and the variable cost. If q represents the number of cones sold in a month, the monthly cost c(q) is given by 
   c(q) = 300 + 0.25q

b) If q cones are sold for $1.25 each, the revenue is given by
   r(q) = 1.25q

c) Profit is the difference between revenue and cost.
   p(q) = r(q) - c(q)
   p(q) = 1.00q - 300 . . . . . . slope-intercept form

d) The equation in part (c) is already in slope-intercept form.
   q - p = 300 . . . . . . . . . . . . standard form
The slope is the profit contribution from the sale of one cone ($1 per cone).
The intercept is the profit (loss) that results if no cones are sold.

e) With a suitable graphing program either form of the equation can be graphed simply by entering it into the program.
   Slope-intercept form. Plot the intercept (-300) and draw a line with the appropriate slope (1).
   Standard form. It is convenient to actually or virtually convert the equation to intercept form and draw a line through the points (0, -300) and (300, 0) where q is on the horizontal axis.

f) Of the three equations created, we presume the one of interest is the profit equation. Its domain is all non-negative values of q. Its range is all values of p that are -300 or more.

g) The x-intercept identified in part (e) is (300, 0). You need to sell 300 cones to break even.

h) Profit numbers are
   425 cones: $125 profit
   550 cones: $250 profit
   700 cones: $400 profit