Question

A company produces 'x' units of geometry boxes in a day. If the raw material of one geometry box costs ₹2 more than the square of the number of boxes produced in a day, the cost of transportation is half the number of boxes produced in a day, and the cost incurred on storage is ₹150 per day. The marginal cost (in ₹) when 70 geometry boxes are produced in a day is:

• A. ₹14,852.50
• B. ₹14,702.50
• C. ₹14,795
• D. ₹5,087.50

Answer 1

The Correct Option is B

The total cost $C(x)$ is the sum of:
Raw material cost: $x(x^2 + 2) = x^3 + 2x$,
Transportation cost: $\frac{5x}{2}$,
Storage cost: 150.
Thus:
$C(x) = x^3 + 2x + \frac{5x}{2} + 150$.
Simplify:
$C(x) = x^3 + \frac{9x}{2} + 150$.
Step 1: Find the marginal cost.
The marginal cost is the derivative of $C(x)$:
$C'(x) = \frac{d}{dx} \left(x^3 + \frac{9x}{2} + 150\right)$.
Differentiate term by term:
$C'(x) = 3x^2 + \frac{9}{2}$.
Step 2: Evaluate at $x = 70$.
Substitute $x = 70$ into $C'(x)$:
$C'(70) = 3(70)^2 + \frac{9}{2}$.
Simplify:
$C'(70) = 3(4900) + \frac{9}{2} = 14700 + 4.5 = 14702.5$.
Final Answer:
14,702.5