Question 1168548
Let's break down this problem step by step.

**Given:**

* Cost function: C(x) = 0.002x³ + 0.1x² + 42x + 300 (where C(x) is in hundreds of dollars)
* Current production: x = 40 security systems

**a) What is the current daily cost?**

* C(40) = 0.002(40)³ + 0.1(40)² + 42(40) + 300
* C(40) = 0.002(64000) + 0.1(1600) + 1680 + 300
* C(40) = 128 + 160 + 1680 + 300
* C(40) = 2268

Since C(x) is in hundreds of dollars, the current daily cost is 2268 * 100 = $226,800.

**b) What would be the additional daily cost of increasing production to 41 security systems daily?**

* C(41) = 0.002(41)³ + 0.1(41)² + 42(41) + 300
* C(41) = 0.002(68921) + 0.1(1681) + 1722 + 300
* C(41) = 137.842 + 168.1 + 1722 + 300
* C(41) = 2327.942

* Additional cost = C(41) - C(40)
* Additional cost = 2327.942 - 2268 = 59.942

Since C(x) is in hundreds of dollars, the additional cost is 59.942 * 100 = $5,994.20.

**c) What is the marginal cost when x = 40?**

* Marginal cost is the derivative of the cost function, C'(x).
* C'(x) = 0.006x² + 0.2x + 42

* C'(40) = 0.006(40)² + 0.2(40) + 42
* C'(40) = 0.006(1600) + 8 + 42
* C'(40) = 9.6 + 8 + 42
* C'(40) = 59.6

Since C(x) is in hundreds of dollars, the marginal cost at x = 40 is 59.6 * 100 = $5,960.

**d) Use marginal cost to estimate the daily cost of increasing production to 42 security systems daily.**

* Marginal cost at x = 40 gives an estimate for the cost of producing one additional unit.
* To estimate the cost of producing two additional units, we can multiply the marginal cost at x = 40 by 2.

* Estimated additional cost for 2 units = 2 * C'(40)
* Estimated additional cost for 2 units = 2 * 59.6 = 119.2

Since C(x) is in hundreds of dollars, the estimated additional cost is 119.2 * 100 = $11,920.

* Estimated cost of producing 42 units:
    * C(42) = 0.002(42)^3 + 0.1(42)^2 + 42(42) + 300
    * C(42) = 0.002(74088) + 0.1(1764) + 1764 + 300
    * C(42) = 148.176 + 176.4 + 1764 + 300
    * C(42) = 2388.576
* C(42) = 2388.576 * 100 = $238,857.60

* C(42) - C(40) = 2388.576 - 2268 = 120.576
* 120.576 * 100 = $12,057.60

The marginal cost estimation of $11,920 is very close to the actual additional cost of $12,057.60.

**Answers:**

a) $226,800
b) $5,994.20
c) $5,960
d) $11,920 (estimated additional cost)