Question

Giving lots of points, I need an answer for all three

8. Profit
The revenue and cost (in dollars) for manufacturing x bicycles are given by the functions
R(x)=-0.04x² +320x and C(x)=85x+120,000.
a. Write a simplified function to represent the profit.


b. What is the profit if 1000 bicycles are produced and sold?


c. How many units must the company produce in order to break-even?

Answer 1

Explanation:

a. The profit function, P(x), can be found by subtracting the cost function from the revenue function.

P(x) = R(x) - C(x)

P(x) = (-0.04x² + 320x) - (85x + 120,000)

P(x) = -0.04x² + 235x - 120,000

b. To find the profit if 1000 bicycles are produced and sold, substitute x=1000 into the profit function:

P(1000) = -0.04(1000)² + 235(1000) - 120,000

P(1000) = -40,000 + 235,000 - 120,000

P(1000) = 75,000

So, the profit would be $75,000 if 1000 bicycles are produced and sold.

c. To find the break-even point, we set the profit function equal to zero and solve for x:

P(x) = 0

-0.04x² + 235x - 120,000 = 0

Using the quadratic formula or factoring, we can find that the break-even point occurs when:

x = (235 +/- sqrt(235^2 - 4(-0.04)(-120,000)))/(2(-0.04))

x = (235 +/- sqrt(55,089,000))/(-0.08)

x = (235 +/- 7,500)/(-0.08)

x = (235 - 7,500)/(-0.08) or (235 + 7,500)/(-0.08)

x = -31,250/ -0.08 or 5,375/-0.08

x = 3906.25 or -64,375

Since negative values of x do not make sense in this context, the break-even point occurs when 3906.25 bicycles are produced and sold.