Word Problem

Question

You work for a small flower shop which wishes to determine how much to charge for a boquet of flowers. The higher price you charge, the fewer bouquets you will sell, but the more money you will make per bouquet. Market research has found that the price P needed to sell a certain number of bouquets per day, x, is P(x) = 50 - 0.5x.

A) If your shop sells x bouquets at a price P(x) how much money will you make from sales in one day? Your answer to this question will have an x in it ( in other words, it will be a function of x). This amount of money is called revenue, and will be denoted by R(x)

B) If the donut shop has a fixed cost of $500 per day for rent and labour and it costs them $10 per bouquet for the flowers and ribbon, find a formula for the cost per day in terms of x, the number of bouquets sold. We will denote this cost function P(x) in terms of x.

C) Use the revenue and cost functions above to find a formula for the profit function P(x) in terms of x.

D) Find the value of x which yields the maximum profit per day. Justify your answer algebraically.

E) Find the price your shop should charge to yield maximum profit.

Answer 1

P=50-x/2

R(x)=Px= 50x-(1/2)x^2

C(x)=500+10x

P(x)=R-C = 50x-(1/2)x^2 - 500 - 10x = 40x -(1/2)x^2-500

maximum profit is found by taking the derivative of the profit function and set = 0

P'(x) = 40 - x = 0

x=40

plug 40 into the profit function to get max profit

max P = 40(40)-(1/2)(40)^2-500 = 1600-800-500= $300