The total revenue function for a product is given by R=475x dollars, and the total cost function for this same product is given by C=10,000+50x+x squared, where C is measured in dollars. For both functions, the input x is the number of units produced and sold.
a. Form the profit function for this product from the two given functions.
b. What is the profit when 15 units are produced and sold?
c. What is the profit when 34 units are produced and sold?
d. How many units must be sold to break even on this product?
a. The profit function is the difference between the total revenue and the total cost:
P(x) = R(x) - C(x)
P(x) = 475x - (10,000 + 50x + x^2)
P(x) = -x^2 + 425x - 10,000
b. To find the profit when 15 units are produced and sold, we substitute x = 15 into the profit function:
P(15) = -(15)^2 + 425(15) - 10,000
P(15) = $3,625
Therefore, the profit when 15 units are produced and sold is $3,625.
c. To find the profit when 34 units are produced and sold, we substitute x = 34 into the profit function:
P(34) = -(34)^2 + 425(34) - 10,000
P(34) = $5,656
Therefore, the profit when 34 units are produced and sold is $5,656.
d. To break even, the profit should be zero. Therefore, we set P(x) = 0 and solve for x:
-x^2 + 425x - 10,000 = 0
Using the quadratic formula, we obtain:
x = (-(425) ± sqrt(425^2 - 4(-1)(-10,000))) / 2(-1)
x = (-(425) ± sqrt(207,025)) / 2
x ≈ 47.44 or x ≈ 210.56
Since x represents the number of units produced and sold, we cannot produce a non-integer number of units. Therefore, we must round up to the nearest integer.
The minimum number of units that must be sold to break even is 48 units.