The telephone company is planning to introduce two new types of executive commun

Question

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 408x+5y consumers will buy the first type and 50+9x7y 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 $2700 per system, what prices x and y will maximize the telephone company's profit?

Explanation / Answer

NOTE: i have solved this problem with cost $1700 and $3000. So replace them with 20 and 27 in your question and then follow the procedure provided. So, your cost will become 20x+27y.

solution-:

cost = 17x + 30y
sales = (40 - 8x + 5y)*x + (50 + 9x - 7y)*y
= 40x - 8x^2 + 5xy + 50y + 9xy - 7y^2
= -8x^2 + 40x + 14xy + 50y - 7y^2
profit = sales - cost
= (-8x^2 + 40x + 14xy + 50y - 7y^2) - (17x + 30y)
= -8x^2 + 23x + 14xy + 20y - 7y^2

p/x = -16x + 14y + 23
p/y = -14y + 14x + 20

Setting them to zeros

16x - 14y = 23 … (1)
14x - 14y = -20 … (2)

Subtracting the (2) from (1)

2x = 43
x = 21.50 (100 dollars)

Plugging this in (2)

14(21.5) - 14y = -20
y = 22.93 (100 dollars)

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