Scenario: You are the manager of a company that manufactures electric chainsaws. Currently the company makes $5,000$ chainsaws each year and sells them for $$200$ each. You suspect that the company should be able to sell more chainsaws and for a higher price. However, if you raise the price too high, not as many would sell. The company also doesn$$t have any storage space so if the company makes more chainsaws than they can sell, they will have to pay someone to store them. Your goal is to maximize profit, that is$,$ the amount of money your company earns minus the amount your company spends.

It costs the company $$95$ for the materials to make each chainsaw, and it costs $$400,000$ each year to run the electric chainsaw factory. You conducted market research and found that at the current price of $$200$ per chainsaw, the company should be able to sell $14,000$ units. You also found that if the price was raised to $$220$ each, the company should be able to sell $11,000$ units.

$1.$ The demand function is the function that gives quantity demanded as a function of price. Assuming demand is linear, find the demand function for these electric chainsaws.

$2.$ Given this demand function, what is the largest number of chainsaws that can be sold?

$3.$ Once you know the demand function and a fixed number of chainsaws is produced, you will set the price as large as possible. What is the revenue function? That is$,$ give the amount of money you make as a function of the quantity produced.

$4.$ Find the profit function.

$5.$ If the current production is $5,000$ chainsaws per year, how much extra profit would the company get if they produced one more chainsaw?

$6.$ Based on the increase in profit for one additional chainsaw, at a level of $5,000,$ would you recommend producing more? Why?

$7.$ If the current production were $20,000$ chainsaws per year, how much extra profit would the company get if they produced one more chainsaw?

$8.$ Based on the increase in profit for one additional chainsaw, at a level of $20,000,$ would you recommend producing more? Why?

$9.$ Explain how analyzing the additional profit of one more chainsaw relates to calculus.

$10.$ Using the previous problems where we analyzed the additional profit of producing one more chainsaw, how would you decide the number of chainsaws to product to maximize profit?

$11.$ The relative rate of change of a function at a point is the rate of change at that point divided by the value of the function at that point. Find the relative rate of change of profit.

$12.$ The elasticity of demand is the negative relative rate of change of change of demand, divided by the relative rate of change of price. Give an intuitive interpretation of what elasticity of demand means.

$13.$ Find the elasticity of demand if the price is $$200.$

$14.$ Find the elasticity of demand if the price is $$220.$