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

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

### 📏 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.

### 🔍 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'.

### 📉 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

### 💡Right-angled triangle

### 💡Rectangle

### 💡Area

### 💡Variable

### 💡Trigonometry

### 💡Tangent

### 💡Derivative

### 💡Optimization

### 💡Vertex

### 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.