A company produces a special new type of TV. The company has fixed costs of $467

Question

A company produces a special new type of TV. The company has fixed costs of $467,000, and it costs $1200 to produce each TV. The company projects that if it charges a price of $2300 for the TV, it will be able to sell 700 TVs. If the company wants to sell 750 TVs, however, it must ower the price to $2000. Assume a linear demand. What price should the company charge to earn a profit of $683,000? It would need to charge (Round answer to nearest dollar. If more than one answer, separate with a comma.)

Explanation / Answer

Fixed costs = $467,000

Cost of production of a TV = $1200

Let x no. of tv's to be sold.

Total cost of producing x tv = 1200 *x + 467,000

Let the projected selling price of a tv be $y

The selling price of x tv's = x*y

Profit = x*y - 1200*x-467,000 = x(y-1200) - 467000

Assuming linear demand equation : y = a*x+C will hold true

Here y = selling price of a tv and x = no. of tv units to be sold, a = slope of equation

When y= 2300, x = 700 => 2300=700a+C

When y=2000, x=750 => 2000 = 750a + C

Solving these two equations:

2300-2000 = 700a-750a => 300 = -50a => a = -6

Hence 2300 = 700*-6+C => 2300+4200 = C => C = 6500

Hence linear demand equation is : y = -6x+6500

=> x = (6500-y)/6

Replacing x in first equation :

Profit = (6500-y)/6 *(y-1200) -467000 = 683000

=> (6500y-7800000-y2+1200y)/6 = 1150000

=> 7700y-7800000-y2 = 6900000

=> y2 -7700y+7800000 = -6900000

=> y2 - 7700y + 147,00,000 = 0

Solving this equation we get :

We get y = 3500, 4200

The company should charge $3500, $4200 to earn a profit of $683000

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