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Question

A computer software company models the profit on its latest video game using the relation

P = –2x^2 + 16x – 30, where x is the number of games produced in hundred thousands and P is the profit in millions of dollars.

a) What are the break-even points for the company? Show your steps.

b) What is the maximum profit that the company can earn? Explain

c) How many games must the company produce to earn the maximum profit? Explain

Answer 1

The break even point is where profit is zero so set the equation equal to zero and solve

-2x^2+16x-30=0. [divide both sides by -2]

x^2 -8x +15=0 [factor]

(x-3)(x-5)=0 so x=3 and x=5 are the solutions. Since these represent hundreds of thousands answer is

300,000 and 500,000 games

The maximum profit is maximum value of p which would be at the vertex since this is a downward facing parabola. So all we need to do to answer the next part is to find the vertex. the x value will represent the number of games and the y value will be the profit

The x coordinate of the vertex is at -b/2a= -16/-4= 4

f(4) = -2(4^2) +16(4) -30=. -32 + 64 -30=2

So Max profit = $2,000,000 producing 400,000 games

Answer 2

Ana

Draw the profit function on Desmos or a graphing calculator.

(a) The company breaks even when the profit = 0. So just find the x-value when the profit is zero from your graph.

(b) Find the highest point on your graph. The y value of that will be the maximum profit.

(c) For this find the x coordinate of the highest point on the graph. Then multiply that by 100,000 to get the number of games.

Answer 3

break even points are when P=0

0=-2x^2 + 16x -30, solve for x

x^2 -8x +15 = 0

(x-5)(x-3) = 0

x = 3, or 5 hundred thousand

P = profit = -2(3)^2 + 16(3) -30 = -18+ 48 -30 = 0, or

P = profit = -2(5)^2 + 16(5) -30 = -50 + 80 - 30 = 0

You might guess max profit is equidistant between x = 3 or 5. x = 4

P(4) = -2(4)^2 + 16(4) -30 = -32 + 64 -30 = 2 million dollars

take the derivative of the profit function and set = 0

P' = -4x +16 = 0

4x = 16

x = 16/4 = 4 hundred thousand gives max profit

P(4) = 2 million dollars, the max profit