Let's solve the problem step by step.

Part A: Exact cost of producing the 21st watch

To find the exact cost of producing the 21st watch, we need to compute the total cost of producing 21 watches and subtract the cost of producing 20 watches. This will give us the difference between the two, which is the cost of producing the 21st watch.

We are given the cost function: $C(x) = 1800 + 60x - 0.8x^2$

1. Total cost for 21 watches: $C(21) = 1800 + 60(21) - 0.8(21^2)$

2. Total cost for 20 watches: $C(20) = 1800 + 60(20) - 0.8(20^2)$

3. Exact cost of the 21st watch: $\text{Exact cost} = C(21) - C(20)$

Let's calculate these values.

Part B: Approximate cost using marginal cost

The marginal cost is the derivative of the total cost function $C(x)$, which gives the rate of change of the cost with respect to the number of watches produced.

The marginal cost function is: $MC(x) = \frac{dC}{dx} = 60 - 1.6x$

To approximate the cost of producing the 21st watch, we substitute $x = 20$ into the marginal cost function, as it gives us the approximate cost of the next watch:

$\text{Approx. cost} = MC(20) = 60 - 1.6(20)$

Now, I'll calculate both parts.### Part A: Exact cost of producing the 21st watch The exact cost of producing the 21st watch is approximately $27.20.

Part B: Approximate cost using marginal cost

The approximate cost of producing the 21st watch, using the marginal cost, is $28.00.

Would you like more details or further clarification?

Here are some related questions:

1. What is the total cost function used in marginal cost analysis?
2. How does marginal cost estimation differ from exact cost calculation?
3. Why is the marginal cost evaluated at $x = 20$ for the 21st watch?
4. How does the second derivative relate to the behavior of the cost function?
5. What is the significance of marginal cost in production economics?

Tip: Marginal cost is an excellent tool for estimating the cost of producing one additional item, but it may not be as accurate for larger increments.