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Question 1186579: Good evening! I sure wish there was a way to submit images on here. I have such a headache trying to figure this out and I really want to understand.
So this problem shows an image of two hexagons stacked. The top hexagon has three triangles kind of like a crown on top. This is what it says: The area of Design B is 5/3 of the area of another design. Make a design that could be the other design. Explain your reasoning.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to approach this problem:
**Understanding the Relationship**
Design B's area is 5/3 *larger* than the area of the unknown design (let's call it Design A). This means Design A is *smaller* than Design B. To find Design A, we need to find a shape whose area is 3/5 of Design B's area.
**Analyzing Design B**
Let's break down Design B into simpler shapes to make calculating its area easier. We can see Design B is composed of:
* Two regular hexagons (one on top of the other)
* Three equilateral triangles
**Making Design A**
The key is to create a design that is visually and proportionally similar to Design B but smaller. Since the area of Design B can be expressed as 5/3 of the area of Design A, Design A will have 3/5 the area of Design B. Here's one way to do it:
1. **Start with one hexagon:** Take *one* of the hexagons from Design B. This will be the base of Design A.
2. **Add proportionally smaller triangles:** The three triangles on top of Design B are equilateral. We need to create three *smaller* equilateral triangles for Design A. A good approach is to make each side of the new equilateral triangle equal to one half the side of the original triangles. In this way, the area of the new triangles is 1/4 of the area of the original triangles.
**Reasoning**
* **Hexagon:** By using only *one* hexagon instead of two, we've reduced the hexagonal area by half.
* **Triangles:** Because the area of a triangle is proportional to the square of its side length, halving the side length makes the area one-quarter of the original triangle. Since Design B has three triangles, Design A will have three triangles each with 1/4 the area of the triangles in design B.
**Why this works (Area Calculation)**
Let 'h' be the area of one hexagon, and 't' be the area of one triangle in Design B. The area of Design B = 2h + 3t.
Design A contains one hexagon and three triangles. The area of the hexagon is h. The area of each triangle is t/4. The area of the three triangles will be 3t/4. The total area of Design A will be h + 3t/4.
If we take the ratio of Design B to Design A we get (2h + 3t)/(h + 3t/4) = 5/3.
**Visual Representation**
It's best to draw this out. Design B looks like two hexagons stacked with a "crown" of three triangles. Design A would look like *one* hexagon with a smaller "crown" of three smaller triangles on top.
This method creates a Design A that is proportionally similar to Design B and has the correct area relationship (3/5 of Design B's area).
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