A start up 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 costs of $100,000, and each computer costs $300 to produce. Find the price that will maximize the company's profit. How many computers will they sell st this price?

To find the price that will maximize the company's profit and the number of computers sold at this price, we can follow these steps:

Step 1: Determine the relationship between the price and the quantity of computers sold.
- We are given that the company can sell 1000 computers if they price them at $500.
- We are also given that the number sold will increase by 10 for every $1 decrease in the price.

Based on this information, we can generate a table to understand the relationship between the price (P) and the quantity of computers sold (Q):

Price (P) | Quantity Sold (Q)
----------------------------
$500 | 1000
$499 | 1010 (increase by 10)
$498 | 1020 (increase by 10)
... and so on

Step 2: Calculate the revenue earned from selling each quantity of computers.
- Revenue is calculated by multiplying the price by the quantity sold. In this case, Revenue (R) = P * Q.

Using the table generated in Step 1, we can calculate the revenue for each price and quantity sold:

Price (P) | Quantity Sold (Q) | Revenue (R)
------------------------------------------
$500 | 1000 | $500,000
$499 | 1010 | $499,999
$498 | 1020 | $497,956
... and so on

Step 3: Calculate the total cost.
- The company has fixed costs of $100,000, and each computer costs $300 to produce.

To calculate the total cost, we multiply the cost per computer by the quantity sold and add the fixed costs:

Total Cost = (Cost per computer * Quantity Sold) + Fixed Costs
= ($300 * Q) + $100,000

Step 4: Calculate the profit.
- Profit is calculated by subtracting the total cost from the revenue. In this case, Profit (Pf) = Revenue (R) - Total Cost.

Using the revenue and total cost calculated in Step 2 and Step 3, we can calculate the profit for each price and quantity sold:

Price (P) | Quantity Sold (Q) | Revenue (R) | Total Cost | Profit (Pf)
--------------------------------------------------------------------------
$500 | 1000 | $500,000 | ($300 * 1000) + $100,000 = $400,000 | $500,000 - $400,000 = $100,000
$499 | 1010 | $499,999 | ($300 * 1010) + $100,000 = $401,000 | $499,999 - $401,000 = $98,999
$498 | 1020 | $497,956 | ($300 * 1020) + $100,000 = $402,000 | $497,956 - $402,000 = $95,956
... and so on

Step 5: Determine the price that maximizes profit.
- Identify the price at which the profit is the highest.

Looking at the profit column in the table generated in Step 4, we can see that the profit is highest at $100,000 when the price is $500. Therefore, the price that will maximize the company's profit is $500.

Step 6: Determine the number of computers sold at the price that maximizes profit.
- Identify the quantity of computers sold at the price that maximizes profit.

From the information given in the problem, we know that when the price is $500, the company can sell 1000 computers. Therefore, the company will sell 1000 computers at the price of $500, which maximizes their profit.

To summarize:
- The price that will maximize the company's profit is $500.
- The company will sell 1000 computers at this price.