Answer to Question #238591 in Calculus for steve

Question

laptop manufacturer has estimated that their pro t in thousands of dollars is given by the

expression 6 x 2 + 30 x 10 where x (in thousands) is the number of units produced. What production range will enable the manufacturer to realize a pro t of at least $ 14,000? Be sure to include a concluding statement in the context of the problem

Answer 1

P(x) = - 6x² + 30x - 10 where x is in thousands of units and P(x) is in thousands of dollars.

 

Solve for x when P(x) = 14

 

- 6x² + 30x - 10 = 14

 

- 6x² + 30x - 24 = 0

 

Divide through by negative 6.

 

x² - 5x + 4 = 0

 

Factor using FOIL.

 

(x - 4)(x - 1) = 0

 

x = 1 and x = 4, but what happens below 1, beyond 4, and between 1 and 4?

 

Because the profit expression is negative in the quadratic term, the parabola opens downward.

 

Because it has two real roots, the vertex is above the x-axis.

 

Profit at or above $14,000 implies that a line at 14 on the profit (vertical) axis would be cut twice by the curve at x = 1 and x = 4.

 

The vertex of the parabola should be above 14. This can be tested with the first term of the quadratic formula: - b/2a.

 

For a = - 6 and b = 30, the x-coordinate of the vertex is - (30)/2(- 6) = 30/12 = 2.5.

 

The value of the function at this value is the corresponding profit (maximum attainable) for this curve.

 

P(2.5) = - 6(6.25) + 30(2.5) - 10

 

P(2.5) = - 37.50 + 75 - 10

 

P(2.5) = 27.50 (Maximum attainable profit is $27,500, which is a point above 14. The parabola is symmetrical, and we expect two points for each line below the maximum attainable profit.)

 

 

That answers part of the question about range of values for x: below 1 and beyond 4, the profit is less than $14,000.

 

Between 1 and 4, the profit is greater than $14,000.

 

The answer, expressed as a compound inequality is

 

1 < x < 4

 

The manufacturer can make from 1000 units up to and including 4000 units to make at least $14,000 of profit