Maxima and Minima Problem No 13 - Application of Derivatives - Diploma Maths - II | Summary and Q&A

TL;DR

Find the dimensions of an open box with a square base to maximize its volume.

Key Insights

• 🙃 The area of the box consists of the base and four sides, which should add up to the given area.
• 😑 By setting dimensions as X, the height can be expressed in terms of X.
• 😥 The volume equation is maximized by finding critical points and determining if they correspond to a maximum using the second derivative.
• 🍱 The dimensions of the box that maximize volume are 8 cm, 8 cm, and 4 cm for length, breadth, and height respectively.

Transcript

click the Bell icon to get latest videos from Ekeeda Hello friends in this video we are going to see one more problem on maxima and minima let us start with problem number 13 a box with square base is to have an open top the area of the material for making the box is 192 square centimeter what should be the dimensions in order that the volume is as... Read More

Questions & Answers

Q: What is the objective of the problem?

The objective is to find the dimensions of an open box's square base to maximize its volume.

Q: How is the area of the box calculated?

The area of the box consists of the five faces: the base (X^2) and four sides (X*Y), where Y is the height. The total area should be equal to 192 square centimeters.

Q: How is the volume of the box calculated?

The volume is calculated by multiplying the length, breadth, and height of the box together.

Q: How is the equation for volume manipulated to find a maximum?

The equation for volume is differentiated with respect to X, and then the derivative is set equal to zero to find critical points.

Q: How is the maximum volume determined?

The second-order derivative is calculated by differentiating the derivative with respect to X again. If the second derivative is greater than zero, it indicates a maximum volume.

Q: What are the dimensions of the box that maximize the volume?

The dimensions of the box that maximize the volume are 8 centimeters for length and breadth, and 4 centimeters for height.

Summary & Key Takeaways

• A box with a square base needs to be made with an open top, and the area of the material for making the box is given as 192 square centimeters.

• The goal is to determine the dimensions of the box that will maximize its volume.

• By considering the side length of the square base as X, the dimensions of the box are found to be 8 centimeters for length and breadth, and 4 centimeters for height.