A certain item sells for $30. If the cost of producing this item is given by C = .05s^3 + 100, find the marginal profit when x = 10.

There is no x in the production cost formula!

C = .05s^3 + 100

I assume you mean "when s=10".

Cost of 10 units
= 0.05(10)³ + 100
=$150

Cost of 11 units
= 0.05(11)³ + 100
= $166.55

So what is the marginal cost, marginal profit?

Using calculus, the marginal cost would be:
C'(s)
= 0.15s²
So
C'(10)
=0.15*10²
=$15

The discrepancy will diminish as the production quantity get larger.

To find the marginal profit, we need to calculate the derivative of the profit function with respect to the number of items produced (s) and evaluate it at s = 10.

The profit function can be given by P(s) = R(s) - C(s), where R(s) represents the revenue and C(s) represents the cost of producing s items.

In this case, the revenue function R(s) is given by R(s) = 30s, as the item sells for $30 and s represents the number of items produced.

The cost function C(s) is given by C(s) = 0.05s^3 + 100.

To find the profit function, we substitute the revenue and cost functions into the profit equation:

P(s) = R(s) - C(s)
= 30s - (0.05s^3 + 100)
= 30s - 0.05s^3 - 100

Now, we need to find the derivative of P(s) with respect to s:

P'(s) = d/ds (30s - 0.05s^3 - 100)
= 30 - 0.15s^2

To find the marginal profit when s = 10, we substitute s = 10 into P'(s):

P'(10) = 30 - 0.15(10)^2
= 30 - 0.15(100)
= 30 - 15
= 15

Therefore, the marginal profit when s = 10 is $15.