AI SL: A thicc boi: Geo+opti+trig (Show that, derivative)(Paper 1 TZ1 May 2021)

Quiroz Math
25 Sept 202214:14

TLDRIn this video, the presenter tackles a geometry problem involving a gift box with a right-angled triangle top. The challenge is to express the area 'a' in terms of the lengths 'p' and 'q' of the sides of an inscribed rectangle. Using trigonometry and algebra, the presenter finds a relationship between 'p' and 'q', derives the formula for 'a', and then calculates its derivative to optimize the area, finding that the minimum area occurs when 'q' equals 8 centimeters.

Takeaways

  • 📏 The problem involves a right-angled triangle and a rectangle inscribed within it, with given lengths of sides PQ and JL.
  • 🎁 The goal is to express the area 'a' of the triangle in terms of 'p' and 'q', which represent the lengths of the sides of the rectangle.
  • 🔍 The area of the triangle is given as 192 square centimeters, which is the area of the top of the gift box.
  • 📐 By using trigonometric properties and the concept of similar triangles, the relationship between 'p' and 'q' is established.
  • 🧩 The formula for the area 'a' is derived as \( a = \frac{3q + 48}{q + 8} \), which simplifies to the given form without 'p'.
  • 📉 To find the derivative of 'a' with respect to 'q', the formula is manipulated to make 'q' the subject, resulting in \( \frac{d a}{d q} = -\frac{192}{q^2} + 3 \).
  • 🔑 The derivative is set to zero to find the value of 'q' that minimizes the area, leading to the equation \( -\frac{192}{q^2} + 3 = 0 \).
  • 🧮 Solving the equation for 'q' gives \( q = 8 \) cm, which is the value that minimizes the area of the gift box's top.
  • 📝 The process involves careful reading of the problem, understanding geometric relationships, and applying calculus for optimization.
  • 📚 The script emphasizes the importance of reading the problem carefully, visualizing geometric shapes, and using trigonometry and algebra to solve.

Q & A

  • What is the shape of the top of the gift box designed by Ellis in the problem?

    -The top of the gift box is in the shape of a right angle triangle, specifically triangle GIK.

  • What are the lengths of the sides GH, JK, and HI of the triangle in the problem?

    -The lengths of GH, JK, and HI are PQ, 8, and 6 respectively.

  • What is the area of the top of the gift box in the problem?

    -The area of the top of the gift box is given as a centimeter squared, but the exact numerical value is not provided in the script.

  • How does the script suggest finding the area of the gift box in terms of P and Q?

    -The script suggests using the formula for the area of a right triangle, which is (base * height) / 2, where the base is 6 + P and the height is Q + 8.

  • What trigonometric concept is used to relate P and Q in the script?

    -The script uses the concept of tangent, which is the ratio of the opposite side to the adjacent side in a right triangle, to relate P and Q.

  • How does the script derive the equation P = 48/Q?

    -The script sets the tangent of two angles equal to each other, resulting in the equation q/6 = 8/p. Cross-multiplying gives 8 * 6 = p * q, which simplifies to p = 48/Q.

  • What is the final expression for the area A of the gift box in terms of Q?

    -The final expression for the area A is A = 192/Q + 3Q + 48.

  • What does the derivative of the area A with respect to Q represent in the context of the problem?

    -The derivative of the area A with respect to Q represents the rate of change of the area with respect to Q, which is used to find the value of Q that minimizes the area.

  • How does the script suggest finding the value of Q that minimizes the area of the gift box?

    -The script suggests setting the derivative of the area A with respect to Q equal to zero and solving for Q to find the value that minimizes the area.

  • What is the value of Q that minimizes the area of the gift box according to the script?

    -The value of Q that minimizes the area of the gift box is 8 centimeters, as found by solving the equation 192/Q^2 + 3 = 0.

Outlines

00:00

📏 Geometry Problem Introduction

The paragraph introduces a geometry problem involving a gift box with a top in the shape of a right-angled triangle. The triangle has a rectangular section inscribed within it, with given lengths PQ=8 and JL=6. The goal is to express the area 'a' of the triangle in terms of 'p' and 'q'. The speaker emphasizes the importance of reading the problem carefully and understanding the given diagram, which includes a right-angled triangle labeled as GIK. The area of the gift box top is given as a centimeter squared, and the task is to find 'a' using the dimensions provided.

05:01

🔍 Analyzing the Geometry and Trigonometry

This paragraph delves into the process of finding the area 'a' of the right-angled triangle using trigonometry and geometric properties. The speaker identifies that the rectangle within the triangle can be used to deduce the lengths of the sides adjacent to the right angle. By comparing the tangents of angles in similar triangles, the speaker sets up an equation to relate 'p' and 'q'. The goal is to express 'p' in terms of 'q' to simplify the area formula. The paragraph concludes with the equation 'P = 48 / Q', which is a key step towards finding an area formula that only contains 'q'.

10:04

📉 Calculus for Optimization

The final paragraph discusses the optimization part of the problem, where Alice wants to minimize the area of the gift box top. The speaker explains the process of taking the derivative of the area formula with respect to 'q' and setting it to zero to find the minimum point, which is a standard procedure in calculus for optimization problems. The derivative is simplified and set to zero, leading to an equation that can be solved for 'q'. The solution 'q = 8' is found, which minimizes the area of the top of the gift box. The speaker also explains the concept of negative exponents and how they are used in the derivative calculation.

Mindmap

Keywords

💡Gift box

A gift box is a container used for presenting gifts. In the video script, the gift box is specifically designed with a top in the shape of a right-angled triangle, which is a key part of the geometric problem discussed. The script mentions that the top of the gift box is made up of a triangle with a rectangular section inscribed inside it, highlighting the geometrical aspect of the gift box's design.

💡Right-angled triangle

A right-angled triangle is a triangle that has one angle that is exactly 90 degrees. In the context of the video, the top of the gift box is described as a right-angled triangle, which is fundamental to the mathematical problem presented. The script uses this geometric shape to introduce the problem of finding the area of the top of the gift box in terms of the given variables P and Q.

💡Rectangle

A rectangle is a quadrilateral with four right angles. In the script, a rectangle is inscribed inside the right-angled triangle of the gift box's top. The dimensions of this rectangle are given as 6 and 8, which are crucial for calculating the area of the triangle and, by extension, the gift box's top.

💡Area

The area is a measure of the two-dimensional space enclosed within a shape. The script's main objective is to find the area of the top of the gift box, which is expressed in square centimeters. The area is calculated using the dimensions of the inscribed rectangle and the lengths of the sides of the triangle, which are related to the variables P and Q.

💡Variable

In mathematics, a variable represents an unknown or changeable value. In the script, P and Q are variables that represent the lengths of certain sides of the geometric shapes involved. The video aims to express the area of the gift box's top in terms of these variables, which is a common approach in algebraic problems.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. The script mentions using trigonometry to solve the problem, specifically the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle. This is used to relate the variables P and Q and to find an expression for the area in terms of a single variable.

💡Tangent

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. In the script, the tangent function is used to establish a relationship between the angles formed by the inscribed rectangle and the sides of the triangle, which helps in deriving an equation that relates P and Q.

💡Derivative

The derivative in calculus is a measure of how a function changes as its input changes. In the context of the video, the derivative of the area function with respect to Q is taken to find the value of Q that minimizes the area of the gift box's top. This is a fundamental concept in optimization, where the derivative is set to zero to find the extremum of a function.

💡Optimization

Optimization is the process of finding the best solution within a set of available options. In the script, the goal is to minimize the area of the gift box's top, which is an optimization problem. The derivative of the area function is used to find the value of Q that achieves this minimum, illustrating a practical application of optimization techniques.

💡Vertex

In the context of a quadratic function, the vertex represents the point at which the function reaches its maximum or minimum value. The script explains that setting the derivative to zero will find the vertex of the quadratic area function, which corresponds to the minimum area of the gift box's top, given the constraints of the problem.

Highlights

Ellis designs a gift box with a top in the shape of a right angle triangle.

The rectangular section H I J L is inscribed inside the triangle.

Lengths of GH, JK, HL, and AJ are PQ, 8, and 6 respectively.

The area of the gift box top is given as a centimeter squared.

The goal is to find 'a' in terms of 'p' and 'Q'.

The top of the gift box is a right angle triangle, not the rectangle.

Using the rectangle's properties to find the base and height for the area calculation.

The area formula is base times height divided by 2.

Relating 'p' and 'Q' through the properties of parallel lines and angles.

Using trigonometry to relate 'p' and 'Q' with tangent of angles.

Equating the tangents to find a relationship between 'p' and 'Q'.

Solving for 'p' in terms of 'Q' to simplify the area formula.

Deriving the area formula to express it solely in terms of 'Q'.

The area 'a' is shown to be 192 divided by 'Q' plus 3 times 'Q' plus 48.

Finding the derivative of 'a' with respect to 'Q' to minimize the area.

Setting the derivative equal to zero to find the minimum area.

Solving the optimization equation to find the value of 'Q'.

The value of 'Q' that minimizes the area is found to be 8 centimeters.

Casual Browsing

AI SL: Paper 1 (TZ1 May 2021)

2024-06-12 14:10:01

AI SL: Paper 1 (TZ2 May 2021)

2024-06-12 15:05:01

Derivative notation and limits [IB Maths AI SL/HL]

2024-06-12 17:45:00

AI SL: Derivatives example 1 (minimum, optimization)

2024-06-12 13:35:01

Scientific Notation [IB Math AI SL/HL]

2024-06-12 13:55:00

Inverse functions [IB Maths AI SL/HL]

2024-06-12 16:00:00