Question

The total cost function for a product is given by C(x)=3x3−9x2−243x+1229, where x is the number of units produced and C is the cost in hundreds of dollars. Use factoring by grouping and then find the number of units that will give a total cost of at least $50,000. Verify the conclusion with a graphing utility.

Answer 1

C(x) = 3x³ − 9x² − 243x + 1229

500 = 3x³ − 9x² − 243x + 1229

Subtract 500 on each side, as follows:

0 = 3x³ − 9x² − 243x + 729

(3x³ − 9x²) - (243x - 729) = 0

3x² (x - 3) - 243 (x - 3) = 0

(x - 3)(3x² - 243) = 0

x = 3, x = 9, x = -9; in the context of this problem, where x is the number

of units produced, negative values of x must be omitted, so x = 3 and x = 9

So we can say that if either 3 units or 9 units are produced the total cost

for the product will be at least $50000

[0, 3] U [9,∞)