Calculus Homework

Question

A company produces a special new type of TV. The company has fixed costs of

​483,000​,

and it costs

​$1100

to produce each TV. The company projects that if it charges a price of

​$2400

for the​ TV, it will be able to sell

750

TVs. If the company wants to sell

800

​TVs, however, it must lower the price to

​$2100.

Assume a linear demand.

What is the maximum profit that can be​ reached?

It is

​$nothing.

​(Round answer to nearest​ cent.)

Answer 1

Told the demand (sales) verses price is linear.

That means if have two points we can find the equation for the equation for the demand verses price.

750 = 2400m + b and 800 = 2100m + b where m is the slope and b a constant.

m = change in demand, divided by change in price. m = (800 - 750)/(2100 - 2400) or m = (80 - 75)/(210 - 240)

m = 5/-30 = -1/6

Now take the value m and solve one of the coordinates for b. 750 = -2400/6 + b yields 750 = -400 + b.

Adding 400 to both sides yields 1150 = b.

Sales = - Price/6 + 1150

The profit equation becomes, -483,000 -1100(1150 - Price/6) + Price(1150 - Price/6) = Profit.

Clearing the parenthesis yields - 483,000- 1,265,000 + 1100xPrice/6 + 1150xPrice - Price^2/6 = Profit.

Multiplying 1150xPrice by 6/6 gives 6900xPrice/6

Combining like terms yields, -1,748,000 + (1100 + 6900)xPrice/6 - Price^2/6 = Profit

or Profit = - Price^2/6 + 8000Price/6 - 1,748,000

There are two ways to find the point of maximum profit. One is to locate the vertex.

The other is to find the point where the slope goes to zero using calculus.

The vertex for a quadratic function = -b/2a, where b is the coefficient of the middle term

and a is the coefficient of the squared term. The vertex occurs at -8000/6 divided by -2/6.

This gives a price of 4000 and sales of 1150 - 4002/6 = 483. As cannot sell one third of a TV.

Profit = -1,748,000 + 8000x4002/6 -4002^2/6 yields -1,748,000 + 8000 x 667 - 4002 x 667

Profit = -1,748,000 + 5,336,000 - 2,669,334 = 918,666

As a check, a price of 3996 give a sales of 1150 - 3996/6 = 484

Profit = -1,748,000 + 8000x3996/6 -3996^2/6 yields -1,748,000 + 8000 x 666 - 3996 x 666

Profit = -1.748,000 + 5,328,000 - 2,661,336 = 918,664