Question

Leah has a home-based business making and selling scented soaps. She initially spent $100 to purchase soap-making equipment, and the materials for each pound of soap cost $5. Leah sells the soap for $15 per pound. Eventually, she will sell enough soap to cover the cost of the equipment. What will be Leah's total sales and costs be? Write a system of equations, graph them, and type the solution.

Answer 1

Equations:
Cost equation C(x) = 100 + 5x
Sales equation S(x) = 15x

Graph provided

Solution at breakeven point i.e. where sales = costs is at x = 10 lbs

Explanation:

Total cost = $100 capital cost of equipment + production cost of $5 per pound of soap

Cost equation C(x) = 100 + 5x where x is the number of pounds of soaps produced

Sales Equation S(x) = 15x where x is the number of pounds of soap sold

The break-even quantity (in lbs) of soap to be sold to pay for equipment is when both equations are equal to each other

100 + 5x = 15x
==> 15x = 100 + 5x
==> 15x - 5x = 100 (subtract 5x on both sides)

==> 10x = 100

==> x = 10 lbs of soap

At this quantity, Sales and Costs are evened out