Question at position 9 A manufacturer of a product has a marginal cost function given by dcdq=0.1q2−20q+1500\frac{dc}{dq}=0.1q^2-20q+1500, where c is the total cost (in dollars) of producing q units of a product. If fixed costs are $30,000, then the total cost of producing 30 units is$66,900.$66,800.$67,000.$66,600.$66,700.

Question

Question at position 9  A manufacturer of a product has a marginal cost function given by dcdq=0.1q2−20q+1500\frac{dc}{dq}=0.1q^2-20q+1500, where c is the total cost (in dollars) of producing q units of a product. If fixed costs are $30,000, then the total cost of producing 30 units is$66,900.$66,800.$67,000.$66,600.$66,700.

BlackTom AI · Accepted Answer

We start by identifying the given information and what is being asked. The marginal cost function is dc/dq = 0.1 q^2 − 20 q + 1500, where c is total cost and q is units produced. Fixed costs are 30,000 dollars. We need the total cost C(30).

Step 1: Integrate the marginal cost to obtain the total cost function up to a constant.
- ∫(0.1 q^2) dq = 0.1 · q^3/3 = (1/30) q^3
- ∫(−20 q) dq = −10 q^2
- ∫(1500) dq = 1500 q
- So C(q) = (1/30) q^3 − 10 q^2 + 1500 q + K, where K is the constant of integration.

Step 2: Use fixed costs to determine the constant K.
- Fixed costs mean C(0) = 30,000.
- C(0) = (1/30)·0^3 − 10·0^2 + 1500·0 + K = K.
- Therefore, K = 30,000.

Step 3: Compute the total cost for q = 30.
- (1/30) q^3 with q = 30: (1/30) · 30^3 = (1/30) · 27,000 = 900.
- −10 q^2 with q = 30: −10 · (30^2) = −10 · 900 = −9,000.
- 1500 q with q = 30: 1500 · 30 = 45,000.
- Now sum: 900 − 9,000 + 45,000 = 36,900.
- Add the fixed cost K = 30,000: 36,900 + 30,000 = 66,900.

Step 4: Compare with the options.
- Option A: $66,900. This matches the calculated total cost C(30) = 66,900, so this option is supported by the math.
- Option B: $66,800. This would require a slightly different calculation or a different constant; the discrepancy here might come from omitting part of the integral or an arithmetic error, but the proper integration plus fixed cost yields 66,900, not 66,800.
- Option C: $67,000. This is higher by 100 than the computed value; possible confusion could stem from rounding or miscomputing one term, but the exact integral result leads to 66,900.
- Option D: $66,600. This is lower by 300; again, not matching the precise integral result with the fixed cost.
- Option E: $66,700. This lies 200 short of the computed value; it would imply subtracting 200 somewhere in the integral or fixed cost, which is not consistent with the given dc/dq and fixed cost.

Additional notes on the reasoning:
- The key step is correctly integrating the marginal cost and then applying the fixed cost as the constant of integration.:
- Double-checking the arithmetic: (1/30)·30^3 = 900, −10·30^2 = −9,000, 1500·30 = 45,000; summing these three terms gives 36,900, and adding 30,000 fixed costs yields 66,900.
- If one were to obtain any of the other numbers, it would indicate a misstep such as using an incorrect antiderivative, forgetting the constant K, or miscomputing a term in the polynomial.

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