a computer company can sell 1000 computers if they price them at $500. The number sold will increase by 10 for every 1$ decrease in the price. The company has fixed cost of $100,000, and each computer costs $300 to produce. Find the price that will maximize the company's profit. How many computers will sell at this price?
let the number of $1 decreases be n
number sold = 1000 + 10n
price of each = 500 - n
profit = (1000 + 10n)(500-n) - 300(1000+10n) - 100000
expand this mess, differentiate it , then set that equal to zero
$102.00
To maximize the company's profit, we need to find the price at which the revenue is maximized and the cost is minimized. Let's break down the problem step by step:
1. Calculate the revenue:
- Revenue = Price x Number of Computers Sold
We can calculate the number of computers sold based on the given information that the number sold will increase by 10 for every $1 decrease in the price. So, the formula for the number of computers sold (N) based on the pricing is:
N = 1000 + (500 - Price) x 10
Note that if the price is less than or equal to $500, the number of computers sold will still be 1000.
2. Calculate the cost:
- The fixed cost is $100,000.
- The cost per computer is $300.
- Total Cost = Fixed Cost + (Cost per Computer x Number of Computers Sold)
3. Calculate the profit:
- Profit = Revenue - Cost
Now, let's find the price that maximizes the profit by trying different prices and calculating the profit for each.
Start by assuming the price is $500, and calculate the number of computers sold, revenue, cost, and profit:
Price = $500
Number of Computers Sold = 1000
Revenue = $500 x 1000 = $500,000
Cost = $100,000 + ($300 x 1000) = $400,000
Profit = $500,000 - $400,000 = $100,000
Next, decrease the price by $1 and re-calculate the number of computers sold, revenue, cost, and profit:
Price = $499
Number of Computers Sold = 1000 + (500 - 499) x 10 = 1010
Revenue = $499 x 1010 = $504,490
Cost = $100,000 + ($300 x 1010) = $403,300
Profit = $504,490 - $403,300 = $101,190
Repeat this process for different prices, decreasing by $1 each time, and calculate the profit for each price until the profit starts to decrease. Once the profit starts to decrease, you will have found the price that maximizes the profit.
Continue this process until you find the price that maximizes the profit. Once you find the price, you can use the formula N = 1000 + (500 - Price) x 10 to calculate the number of computers sold at that price.