Expression for combined area of triangle and square | Differential Calculus | Khan Academy

TL;DR

Given a 100 meter wire, a cut is made to construct an equilateral triangle and a square. The problem is to find the optimal cut location that minimizes the combined areas of the triangle and square.

Transcript

Let's say that I have a 100 meter long wire. So that is my wire right over there. And it is 100 meters. And I'm going to make a cut someplace on this wire. And so let's say I make the cut right over there. With the left section of wire-- I'm going to obviously cut it in two-- with the left section, I'm going to construct an equilateral triangle. An... Read More

Key Insights

• 💇 The problem involves finding the optimal cut location on a wire to minimize the combined areas of an equilateral triangle and a square.
• 💇 The dimensions of the triangle and square are determined by the cut location.
• 🙃 The area of an equilateral triangle can be expressed as (sqrt(3)s^2)/4, where s is the length of its sides.
• 😑 The combined area can be expressed as a function of the cut location.
• 💇 The objective is to minimize the combined area by determining the optimal cut location.
• ❓ The formula for the combined area is (sqrt(3)x^2)/36 + (100 - x/4)^2.
• 💇 To find the optimal cut location, the combined area function needs to be minimized.

Questions & Answers

Q: What is the objective of this problem?

The objective is to find the location to make the cut that minimizes the combined areas of the equilateral triangle and square.

Q: How are the dimensions of the triangle and square determined?

The length of the triangle's sides is x/3, while the length of the square's sides is 100 - x/4.

Q: What is the expression for the area of an equilateral triangle?

The area of an equilateral triangle with side length s can be expressed as (sqrt(3)s^2)/4.

Q: How can the combined area of the triangle and square be expressed as a function of the cut location?

The combined area is given by the formula (sqrt(3)x^2)/36 + (100 - x/4)^2.

Summary & Key Takeaways

• The problem involves making a cut on a 100 meter wire to construct an equilateral triangle and a square.

• The length of the triangle's sides is x/3, and the length of the square's sides is 100 - x/4.

• The area of an equilateral triangle can be expressed as (sqrt(3)s^2)/4, while the area of the square is (100 - x/4)^2.