the telephone company is planning to introduce two new types of executive communications systems that it hopes to sell to its largest commercial customers. It is estimated that if the first type of system is priced at x hundred dollars per system and the second type at y hundred dollars per system, approximately 6-3x+2y consumers will buy the first type and 70+3x-5y will buy the second type. If the cost of manufacturing the first type is $2000 per system and the cost of manufacturing the second type is $2000 per system, what prices x and y will maximize the telephone company's profit?

To find the prices that will maximize the telephone company's profit, we need to formulate an expression for the profit and then optimize it using calculus.

Let's start by defining the profit function. Profit is calculated by subtracting the cost of manufacturing from the revenue generated by selling the systems.

For the first type of system, the revenue is given by (6 - 3x + 2y) * x hundred dollars since 6 - 3x + 2y represents the number of consumers who will buy it.

For the second type of system, the revenue is given by (70 + 3x - 5y) * y hundred dollars since 70 + 3x - 5y represents the number of consumers who will buy it.

Now we can express the profit function as follows:

Profit(x, y) = (6 - 3x + 2y) * x - (2000 * x) + (70 + 3x - 5y) * y - (2000 * y)

To maximize the profit, we need to find the values of x and y that yield the maximum value for the profit function. We can achieve this by taking partial derivatives of the profit function with respect to both x and y and then setting them equal to zero.

∂Profit/∂x = 0 and ∂Profit/∂y = 0

Next, solve these equations simultaneously to find the values of x and y that maximize the profit. The resulting values will give you the optimal prices for the two types of executive communication systems.