It seems that every year the price of Christmas trees increases and it peaks a week before Christmas Eve (December 17th). You contacted a tree grower from South Carolina and scored a good deal with him. He said the price per tree depends on the time of year you buy it. He agreed on the following formula for the price:p(n)=0.02 (n-40)^2, where is the number of days before December 17th and p(n) is the price per tree including transportation to your location. You estimate that the storage cost at your facility is going to be $0.56 per tree per day. The two parts below are independent. Part 1) Let’s assume that you will sell all the trees for the same price on December 17th (n=0). Create a function, C(n)that represents your total cost for buying a tree and storing it until you sell it on December 17th (n=0) and graph the function. Find the relative minimum of this function using derivatives and show your work. What is the selling price per tree if you want to have 20% markup on cost?
It seems that every year the price of Christmas trees increases and it peaks a week before Christmas Eve (December 17th). You contacted a tree grower from South Carolina and scored a good deal with him. He said the price per tree depends on the time of year you buy it. He agreed on the following formula for the price:p(n)=0.02 (n-40)^2, where is the number of days before December 17th and p(n) is the price per tree including transportation to your location. You estimate that the storage cost at your facility is going to be $0.56 per tree per day.
The two parts below are independent.
Part 1)
Let’s assume that you will sell all the trees for the same price on December 17th (n=0).
- Create a function, C(n)that represents your total cost for buying a tree and storing it until you sell it on December 17th (n=0) and graph the function.
- Find the relative minimum of this function using derivatives and show your work.
- What is the selling price per tree if you want to have 20% markup on cost?
Part 2)
It’s unlikely that you will sell all the trees on the same day but rather you will be selling them slowly and increase the price as Christmas approaches. Let’s assume that the final average selling price per tree you achieve if you will be selling it for days before December 17th is s(n)=-e^(-0.15n+2.5) -n+60.
- Graph the function s(n) and explain why the graph looks like this.
- Assume that you have to pay for storing all the trees you buy and the cost is $0.56 per tree per day.
- Create a function, P(n), that represents your profit per tree, i.e., the function is the difference of the average tree selling price and your total cost for buying a tree days before December 17th and storing it.
- Use derivatives to find the maximum of P(n) and determine when you should buy trees to sell.
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