Basic Applications of CalculusPYQ Jan 26Question 4515 of 28
All Questions

The cost of production of an item is given as C=50x−5x2+x36\displaystyle C=50x - 5x^2 + \frac{x^3}{6} where 'x' is number of items to be produced. If the average cost and marginal cost are equal then, what quantity of items should be produced?

Options

A20
B15
C10
D5
For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option b — 15

All Options:

  • A20
  • B15
  • C10
  • D5

Detailed Solution & Explanation

Given the cost function:
C(x)=50x−5x2+x36C(x) = 50x - 5x^2 + \frac{x^3}{6}
where x\displaystyle x represents the quantity of items produced.

**Step 1: Find the Average Cost (AC\displaystyle AC)**
The Average Cost is the total cost divided by the quantity x\displaystyle x:
AC=C(x)x=50x−5x2+x36x=50−5x+x26AC = \frac{C(x)}{x} = \frac{50x - 5x^2 + \frac{x^3}{6}}{x} = 50 - 5x + \frac{x^2}{6}

**Step 2: Find the Marginal Cost (MC\displaystyle MC)**
The Marginal Cost is the derivative of the total cost function with respect to x\displaystyle x:
MC=dCdx=ddx(50x−5x2+x36)MC = \frac{dC}{dx} = \frac{d}{dx}\left(50x - 5x^2 + \frac{x^3}{6}\right)MC=50−10x+3x26=50−10x+x22MC = 50 - 10x + \frac{3x^2}{6} = 50 - 10x + \frac{x^2}{2}

**Step 3: Equate Average Cost and Marginal Cost (AC=MC\displaystyle AC = MC)**
50−5x+x26=50−10x+x2250 - 5x + \frac{x^2}{6} = 50 - 10x + \frac{x^2}{2}
Subtract 50\displaystyle 50 from both sides:
−5x+x26=−10x+x22-5x + \frac{x^2}{6} = -10x + \frac{x^2}{2}
Add 10x\displaystyle 10x to both sides:
5x=x22−x265x = \frac{x^2}{2} - \frac{x^2}{6}
Find a common denominator for the right side:
5x=x2(3−16)5x = x^2 \left(\frac{3 - 1}{6}\right)5x=2x26=x235x = \frac{2x^2}{6} = \frac{x^2}{3}
Assuming x≠0\displaystyle x \neq 0 (production is positive), we can divide both sides by x\displaystyle x:
5=x3  ⟹  x=155 = \frac{x}{3} \implies x = 15
Thus, the quantity of items to be produced is 15\displaystyle 15.

Hence, **Option B** is the correct answer.

About This Chapter: Basic Applications of Calculus

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Limits, Continuity, Derivatives, Integrals

This chapter covers Limits, Continuity, Derivatives, Integrals and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

More Questions from Basic Applications of Calculus

The lim⁡x→2x2−4x+4x−2=\displaystyle \lim_{x \to 2} \frac{x^2-4x+4}{x-2} =

Find the value of lim⁡x→4x2−2x−8x−4\displaystyle \lim_{x \to 4} \frac{x^2 - 2x - 8}{x - 4}

Evaluate: ∫24(3x−2)2dx\displaystyle \int_2^4 (3x - 2)^2 dx

Determine f(x)\displaystyle f(x), given that f′(x)=12x2−4x\displaystyle f'(x) = 12x^2 - 4x and f(−3)=17\displaystyle f(-3) = 17

Find dydx\displaystyle \frac{dy}{dx} where x=et+e−t2\displaystyle x = \frac{e^t + e^{-t}}{2} and y=et−e−t2\displaystyle y = \frac{e^t - e^{-t}}{2}

What is the differential function of x2+2\displaystyle \sqrt{x^2 + 2} ?

Ready to Master Basic Applications of Calculus?

Practice all 28 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free