Question

Your company manufactures automobile alternators, and production is partially automated through the use of robots. In order to meet production deadlines, your company calculates that the numbers of laborers and robots must satisfy the constraint

xy = 10,000,


where x is the number of laborers and y is the number of robots. Your company currently uses 400 robots and is increasing robot deployment at a rate of 16 per month. How fast is it laying off laborers?

Answer 1

The company, using the equation xy = 10,000, is increasing robots at 16 per month and calculating the rate of laying off laborers. By differentiation, the rate is found to be 256 laborers laid off per month.

Step-by-step explanation:

The company's production constraint is given by the equation xy = 10,000, where x is the number of laborers and y is the number of robots. If the company is using 400 robots (y) and increases the number of robots at a rate of 16 per month (dy/dt), we need to find how fast it is laying off laborers (dx/dt).

To solve for dx/dt, we first differentiate both sides of the equation with respect to time (t), assuming x and y are functions of t. We get x(dy/dt) + y(dx/dt) = 0. With y = 400 and dy/dt = 16, we solve for dx/dt. The equation becomes 400*(16) + x*(dx/dt) = 0, which simplifies to dx/dt = -6400/x.

With the initial condition of 400 robots, the number of laborers needed x can be found by x = 10,000 / 400 = 25. Substituting this into our derivative equation yields dx/dt = -6400/25 = -256. Therefore, the company is laying off laborers at a rate of 256 per month.