A regular square pyramid with base side 20 cm and height 45 cm is melted and recast into several identical regular triangular pyramids. Each smaller pyramid has an equilateral triangular base of side 10 cm and height 10√3 cm. How many such regular triangular pyramids can be formed?

Introduction / Context:
This is a volume conservation problem involving pyramids. A large regular square pyramid is melted and recast into smaller regular triangular pyramids. Since melting and recasting do not change the total volume (ignoring any loss), the sum of the volumes of the smaller pyramids must equal the volume of the original pyramid. The question checks your ability to use the volume formula for pyramids and perform accurate geometric calculations.

Given Data / Assumptions:

• Original solid: regular square pyramid with base side 20 cm and height 45 cm.
• Smaller solids: regular triangular pyramids with equilateral base of side 10 cm and height 10√3 cm.
• All smaller pyramids are identical.
• Volume is conserved during melting and recasting.
• We need the number of smaller pyramids produced.

Concept / Approach:
The volume of any pyramid (square based or triangular based) is given by V = (1/3) * base area * height. We first calculate the volume of the original square pyramid. Then we compute the volume of one triangular pyramid using the area formula for an equilateral triangle. Finally, we divide the original volume by the volume of one small pyramid to obtain the number of pyramids formed.

Step-by-Step Solution:
Step 1: Compute the base area of the square pyramid. Base side = 20 cm, so base area A₁ = 20 * 20 = 400 cm^2. Step 2: Height of the square pyramid is h₁ = 45 cm. Volume of the square pyramid V₁ = (1/3) * A₁ * h₁ = (1/3) * 400 * 45 = 400 * 15 = 6000 cm^3. Step 3: For one triangular pyramid, the base is an equilateral triangle of side 10 cm. Step 4: Area of an equilateral triangle with side a is A = (√3 / 4) * a^2. Here a = 10 cm, so base area A₂ = (√3 / 4) * 100 = 25√3 cm^2. Step 5: Height of each triangular pyramid is h₂ = 10√3 cm. Volume of one triangular pyramid V₂ = (1/3) * A₂ * h₂. Step 6: Substitute A₂ and h₂: V₂ = (1/3) * 25√3 * 10√3 = (1/3) * 25 * 10 * (√3 * √3) = (1/3) * 250 * 3 = 250 cm^3. Step 7: Since volume is conserved, number of smaller pyramids n = V₁ / V₂ = 6000 / 250 = 24.

Verification / Alternative check:
You can verify by multiplying the number of small pyramids by the volume of one. If there are 24 pyramids, total volume = 24 * 250 = 6000 cm^3, which equals the original volume of the square pyramid. This confirms that no arithmetic mistakes were made and that exactly 24 pyramids can be formed.

Why Other Options Are Wrong:
If the answer were 20, the total volume would be 20 * 250 = 5000 cm^3, which is less than the original volume, violating volume conservation. Similarly, 27 pyramids would give 6750 cm^3, and 28 pyramids would give 7000 cm^3, both larger than the available volume. Only 24 gives a perfect match with the original pyramid volume.

Common Pitfalls:
Some learners forget that the base areas are different (square versus equilateral triangle) and mistakenly use the wrong area formula. Others may incorrectly compute the area of an equilateral triangle or forget that √3 * √3 = 3. Another pitfall is to misapply the pyramid volume formula and leave out the factor 1/3. Working methodically and writing each formula clearly helps avoid these issues.

Final Answer:
The number of regular triangular pyramids formed is 24.