Applications of Derivatives Case Study Questions Class 12 Maths Chapter 6

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Last Updated on July 21, 2025 by XAM CONTENT

Hello students, we are providing case study questions for class Class 12 Maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are very less. So, to help students we have created chapterwise case study questions for class Class 12 Maths. In this article, you will find case study questions for cbse class Class 12 Maths chapter 6 Applications of Derivatives.

Chapter	Applications of Derivatives
Type of Questions	Case Study Questions
Nature of Questions	Competency Based Questions
Board	CBSE
Class	Class 12
Subject	Maths
Useful for	Class 12 Studying Students
Answers provided	Yes
Difficulty level	Mentioned
Important Link	Class 12 Maths Chapterwise Case Study

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Table of Contents

Case Study Questions on Applications of Derivatives

Case Study:
A company manufactures and sells cylindrical cans with fixed volume 500 cm³. The material for the top and bottom of the can costs twice as much per cm² as the material for the side. Let the radius of the base be r cm and the height be h cm.

The total cost C of making a can is given by:

\[ C = 2 \times \text{(Cost of top and bottom)} + \text{(Cost of side)} \] \[ = 2 \times 2\pi r^2 k + 2\pi r h k = 4\pi r^2 k + 2\pi r h k \]

where \(k\) is the cost per cm² for the side material.

Since the volume is fixed:

\[ \pi r^2 h = 500 \implies h = \frac{500}{\pi r^2} \]

Questions:

Express C in terms of r only, and show all working.
Find the value of r for which the cost C is minimum (state whether it is a minimum).
Give a real-life interpretation of the result.

Solutions:

1. Expressing C in terms of r:

\[ C = 4\pi r^2 k + 2\pi r h k \] Substituting \( h = \frac{500}{\pi r^2} \): \[ C = 4\pi r^2 k + 2\pi r \left( \frac{500}{\pi r^2} \right) k = 4\pi r^2 k + \frac{1000k}{r} \]

2. Finding the value of r for minimum cost:

Differentiate \( C \) with respect to \( r \): \[ \frac{dC}{dr} = 4\pi \cdot 2 r k – 1000k r^{-2} = 8\pi r k – \frac{1000k}{r^2} \] Set derivative to zero: \[ 8\pi r k – \frac{1000k}{r^2} = 0 \implies 8\pi r^3 = 1000 \implies r^3 = \frac{1000}{8\pi} \implies r = \left( \frac{125}{\pi} \right)^{1/3} \] To check for a minimum, observe that the second derivative at this value is positive (since \( r > 0 \)), confirming a minimum.

3. Real-life interpretation:

To minimize total manufacturing cost for a can of specified volume, the radius should be \( r = \left( \frac{125}{\pi} \right)^{1/3} \) cm.
This ensures minimal use of expensive materials for the top/bottom relative to the side, optimizing manufacturing costs for the fixed volume product.

We hope the given case study questions for Applications of Derivatives Class Class 12 helps you in your learning.

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Topics from which case study questions may be asked

Rate of Change of Quantities
Increasing and Decreasing Functions
Maxima and Minima (First and Second Derivative Test)
Tangents and Normals
Simple Problems on Maxima and Minima

Using first and second derivative tests to find maxima and minima helps solve real-world optimization problems in physics, economics, and engineering.

Frequently Asked Questions (FAQs) on Applications of Derivatives Case Study Questions

Q1: What is a case study question in mathematics?

A1: A case study question in mathematics is a problem or set of problems based on a real-life scenario or application. It requires students to apply their understanding of mathematical concepts to analyze, interpret, and solve the given situation.

Q2: How should students tackle case study questions in exams?

A2: To tackle case study questions effectively, students should:
Read the problem carefully: Understand the scenario and identify the mathematical concepts involved.
Break down the problem: Divide the case study into smaller parts to manage the information better.
Apply relevant formulas and theorems: Use the appropriate mathematical tools to solve each part of the problem.

Q3: Why are case study questions included in the Class 12 Maths curriculum?

A3: Case study questions are included to bridge the gap between theoretical knowledge and practical application. They help students see the relevance of what they are learning and prepare them for real-life situations where they may need to use these mathematical concepts.

Q4: Are there any online resources or tools available for practicing Applications of Derivatives case study questions?

A12: We provide case study questions for CBSE Class 12 Maths on our website. Students can visit the website and practice sufficient case study questions and prepare for their exams. If you need more case study questions, then you can visit Physics Gurukul website. they are having a large collection of case study questions for all classes.