Question

A company produces two types of bicycles, A and B. The cost of producing each type is given by the two functions CA (x) = 0.05x2 + 50x + 1000 and CB (x) = 0.03x2 + 70х + 2000, where x represents the number of hours worked on the assembly line. Nicole is asked to determine the number of hours worked on the assembly line where the cost functions are the same. Determine if Nicole's work is correct or incorrect and justify your decision. For any portion of Nicole's response in which there is an error, provide and explain an alternative response.

Answer 1

The cost of producing bicycles A and B is the same when x is approximately -47.72 hours or 1082.73 hours.

Nicole's work is correct. The cost of producing the two bicycles is the same when the two cost functions are equal.

This means that we need to find the values of x for which CA(x) = CB(x).

We can solve this equation by setting the two functions equal to each other and solving for x:

0.05x^2 + 50x + 1000 = 0.03x^2 + 70x + 2000

Simplifying the equation, we get:

0.02x^2 - 20x - 1000 = 0

We can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation. In this case, a = 0.02, b = -20, and c = -1000.

Substituting these values into the formula, we get:

x = (20 ± √((-20)^2 - 4 * 0.02 * -1000)) / (2 * 0.02)

x = (20 ± √(440)) / 0.04

x = (20 ± 20.41) / 0.04

x ≈ -47.72 or x ≈ 1082.73

Therefore, the cost of producing bicycles A and B is the same when x is approximately -47.72 hours or 1082.73 hours.