A popular designer purse sells for $500 and 45,000 purses are sold each month. The marketing specialists have done some research and discovered that for each $20 decrease in price, they can sell 5,000 more purses each month. How much should the company charge for the purse so they can maximize their monthly revenue? (Hint: let x be the number of $20 decreases)
To determine the price at which the company can maximize its monthly revenue, we need to find the value of x (the number of $20 decreases).
Let's start by determining the increase in the number of purses sold for each $20 decrease. According to the information given, a $20 decrease equals 5,000 more purses sold each month.
To find the total number of purses sold when the price decreases by $20, we can calculate: 45,000 + (5,000 * x), where x is the number of $20 decreases.
Now, let's determine the price at which the company should charge for the purse in order to maximize their monthly revenue. We know that the selling price of the purse is initially $500.
For each $20 decrease in price, the revenue earned from each purse sold would be ($500 - $20) = $480.
Thus, the revenue generated from selling 45,000 purses initially is: 45,000 * $500 = $22,500,000.
Now, if the price decreases by $20, the revenue generated from selling the additional 5,000 purses would be: 5,000 * $480 = $2,400,000.
Therefore, the total revenue for x number of $20 decreases in price would be: $22,500,000 + ($2,400,000 * x).
To maximize the monthly revenue, we need to find the value of x that maximizes the total revenue expression.
Since the total revenue is a linear function of x, it will be maximized at the highest possible x value. In this case, we can divide the total revenue by $2,400,000 and round it up to the nearest whole number to get the value of x.
Thus, x = ceil($22,500,000 / $2,400,000) = ceil(9.375) = 10.
Therefore, the company should decrease the price by $20 ten times to maximize its monthly revenue.
To find the price at which the company should charge, we can calculate: $500 - ($20 * 10) = $300.
Hence, the company should charge $300 for the purse to maximize their monthly revenue.
revenue = (500 - 20 x)(45000 + 5000 x)
... = 22500000 +1600000 x - 100000 x^2
xmax = -1600000 / (2 * -100000)