Page 3: Problem-Solving Steps

This page provides a detailed breakdown of the steps to solve part $a$ of the Higher Maths Optimisation question. It demonstrates how to derive the surface area formula for the novelty chocolate box.

Key steps in the solution process:

1. Visualize the box dimensions and identify component parts
2. Calculate areas of the outer and inner squares
3. Determine the volume of the box and tunnel
4. Express the height $h$ in terms of x using the given volume
5. Calculate the surface area of each component (sides, top/bottom, tunnel)
6. Combine all surface area components to derive the final formula

Example: The surface area of the sides is calculated as 4 * $3x * h$, where 3x is the side length and h is the height.

Vocabulary: Cuboid - a three-dimensional rectangular box-shaped object.

Definition: Surface Area - the total area of all surfaces of a three-dimensional object.

Page 4: Optimization Solution

This page focuses on solving part $b$ of the question, which involves finding the minimum value of the surface area A. This section demonstrates key techniques in Higher Maths Differentiation and Optimization.

Steps to find the minimum value of A:

1. Start with the equation A = 16x² + 4000/x
2. Find the first derivative (A') and set it equal to zero
3. Solve for x to find stationary points
4. Calculate the second derivative (A") to determine the nature of stationary points
5. Substitute the found x-value back into the original equation to find the minimum A

Highlight: The solution uses the second derivative test to confirm that the stationary point is indeed a minimum, which is crucial in Higher Maths Optimisation questions and Answers.

Example: The minimum value of A is found to be 1200 cm² when x = ∛125 ≈ 5 cm.

This detailed solution provides valuable insight into solving Higher Maths differentiation optimization questions, particularly those involving surface area minimization in practical contexts.

Page 2: Problem Introduction

This page introduces a Higher Maths Optimisation Past Paper question from 2019 about minimizing the surface area of a novelty chocolate box. The problem provides specific dimensions and constraints for the box design.

Key details of the problem:

• The box is a cuboid with a cuboid-shaped tunnel through it
• Height of the box is h centimeters
• Top of the box is a square with side 3x centimeters
• End of the tunnel is a square with side x centimeters
• Volume of the box is 2000 cm³

The question asks students to: a) Show that the total surface area A cm² of the box is given by A = 16x² + 4000/x b) Find the minimum value of A to minimize production costs

Highlight: This question combines concepts of geometry, algebra, and calculus, making it an excellent example of Higher Maths Differentiation questions and Answers.