Problems 5 through 7: Cost and Revenue In manufacturing, the cost of production may increase as a business tries to scale to larger volumes of product – the business has more employees to pay, needs to buy more equipment, and may need to buy or lease more facilities to handle the mass production.  In some cases, the relationship between cost and quantity can be a quadratic function.  Suppose that a small business owner makes a product with the following monthly cost function: 
𝐶(𝑞) = 4𝑞 +

Here, C represents the cost of producing q many units of the product.  In addition, market research suggests that the following linear function can be used to predict the monthly quantity demanded based on price p: 
𝐷(𝑝) = 20 − 0.5𝑝 
Problem 5 (Related Exercises: Section 1.2 #7 and Textbook Section 1.2 Example 1) Since D(p) represents the demanded quantity of the product, we can let q=D(p).  In doing so, we can consider the composite function: 
𝐶(𝐷(𝑝)) 
Interpret the meaning of this function.  What does it represent in the context of business? 
Problem 6 (Related Exercises: Section 1.5 #7 and 8) Revenue is the amount of money a business makes from selling their product.  If a business makes a quantity q of their product and sells them all at a price of p, the total revenue would be: 
𝑅 = 𝑞 ⋅ 𝑝 
To ensure that all the product inventory is sold, the business should make enough of the product to meet the demand function; thus, the monthly revenue as a function of price becomes: 
𝑅(𝑝) = 𝐷(𝑝) ⋅ 𝑝 
For our specific example, the revenue function would be: 
𝑅(𝑝) = (20 − 0.5𝑝) ⋅ 𝑝 
Find the maximum revenue.  At what price should the business sell the product to achieve this revenue? 
Problem 7 (Requires algebra and factoring; graphing might also help) The break-even point is the point at which cost of production and revenue are equal.  Mathematically, this can be represented: 
𝐶(𝐷(𝑝)) = 𝑅(𝑝) 
Here, we are considering both cost and revenue as functions of the selling price.  What is the minimum selling price needed break-even?