Question

The cost C (in dollars) of manufacturing a number of high-quality computer laser printers is C(x) = 15x4/3 + 15x2/3 + 650,000 Currently, the level of production is 729 printers and that level is increasing at the rate of 300 printers per month. Find the rate at which the cost is increasing each month. The cost is increasing at about $ per month TIP Enter your answer as an integer or decimal number. Examples: 3,-4,5.5172 Enter DNE for Does Not Exist, oo for Infinity Get Help: Video eBook

Answer 1

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.

Answer 2

The rate at which the cost is increasing each month is approximately $45.637.

Step-by-step explanation:

To find the rate at which the cost is increasing each month, we need to find the derivative of the cost function, C(x). The cost function is given as C(x) = 15x4/3 + 15x2/3 + 650,000. Taking the derivative of C(x) will give us the rate of change of the cost with respect to the number of printers produced. Using the power rule, the derivative of C(x) is dC/dx = 20x1/3 + 10x-1/3.

Now we know that the level of production is 729 printers and is increasing at the rate of 300 printers per month. Plugging these values into the derivative, we get dC/dx = 20(729)1/3 + 10(729)-1/3 ≈ 45.637. Therefore, the cost is increasing at a rate of approximately $45.637 per month.