Final answer:
To find the rate at which the cost is increasing each month, differentiate the cost function with respect to time and substitute the given values. The rate at which the cost is increasing is approximately $200.848 per month.
Step-by-step explanation:
To find the rate at which the cost is increasing each month, we need to differentiate the cost function with respect to time. In this case, the number of printers produced is increasing at a rate of 300 printers per month, so we can substitute dx/dt = 300 into the derivative of the cost function. The derivative of the cost function, C'(x), will give us the rate at which the cost is increasing with respect to the number of printers.
First, let's find C'(x) by differentiating C(x) using the power rule.
C'(x) = (4/3) * 15x^(4/3 - 1) + (2/3) * 15x^(2/3 - 1)
Simplifying, we have C'(x) = 20x^(1/3) + 10x^(-1/3). Now, we substitute dx/dt = 300 and x = 729 into C'(x).
C'(729) = 20 * 729^(1/3) + 10 * 729^(-1/3).
Using a calculator, we can find that C'(729) ≈ 200.848.