suppose that the total cost function for a manufacturer is C=100+X^2
where x represents the weekly production of widgets. if production is increasing at the rate of 500 calculators per week, when production output is 6000 calculators. find the rate of increase in cost.
dx/dt = 500
dc/dx = 0 + 2 x
dc/dt = dc/dx * dx/dt
= 2 x * 500
so at x = 6000
dc/dt = 12,000 * 500
6000000
To find the rate of increase in cost, we need to calculate the derivative of the cost function with respect to production, and then substitute the given values.
Given:
Cost function: C(x) = 100 + x^2
Rate of production increase: 500 widgets/week
Production output: 6000 widgets
1. Calculate the derivative of the cost function with respect to x:
C'(x) = dC/dx = 2x
2. Substitute the given production output value (x = 6000):
C'(6000) = 2 * 6000 = 12000
The rate of increase in cost when production output is 6000 calculators is 12000 units of cost per change in production.