Question 1168548: 3. Marginal cost. Suppose the daily cost, in hundreds of
dollars, of producing x security systems is
C1x2 = 0.002x3 + 0.1x2 + 42x + 300,
and currently 40 security systems are produced daily.
a) What is the current daily cost?
b) What would be the additional daily cost of increasing
production to 41 security systems daily?
c) What is the marginal cost when x = 40?
d) Use marginal cost to estimate the daily cost of increasing
production to 42 security systems daily.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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)
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