To solve this problem, let's analyze the given information and determine the volume of each pyramid in the sequence.

Step 1: Understanding the Volume Formula

The volume $V$ of a pyramid with a square base of side length $s$ and height $h$ is given by: $V = \frac{1}{3} \times s^2 \times h$

Step 2: Initial Pyramid Information

For the first pyramid:

• Volume: 200 cubic units
• Height: $h = \frac{3}{4} s$, where $s$ is the side length of the base.

Using the volume formula for the first pyramid, we substitute $h = \frac{3}{4} s$: $200 = \frac{1}{3} \times s^2 \times \frac{3}{4} s$ $200 = \frac{1}{4} s^3$ Solving for $s$: $s^3 = 800 \Rightarrow s = \sqrt[3]{800} = 8.9282 \, (\text{approx.})$

Step 3: Volume Reduction in Sequence

Each subsequent pyramid has a base area that is half of the previous pyramid’s base area. Since the area of a square is $s^2$, this means each new side length $s$ will reduce by a factor of $\frac{1}{\sqrt{2}}$.

Thus, each pyramid in the sequence has a volume that is half of the previous pyramid’s volume.

Step 4: Volume Sequence and Total Volume

Since each pyramid’s volume is half of the one before it, this forms a geometric sequence where:

• First term $V_1 = 200$ cubic units
• Common ratio $r = \frac{1}{2}$

The sum of the first $n$ terms of a geometric sequence is: $S_n = V_1 \frac{1 - r^n}{1 - r}$

For the first 5 pyramids: $S_5 = 200 \times \frac{1 - \left( \frac{1}{2} \right)^5}{1 - \frac{1}{2}}$ $S_5 = 200 \times \frac{1 - \frac{1}{32}}{\frac{1}{2}}$ $S_5 = 200 \times \frac{\frac{31}{32}}{\frac{1}{2}}$ $S_5 = 200 \times \frac{31}{16} = 387.5$

Final Answer

The total volume of the first 5 pyramids is 387.5 cubic units.

Would you like more details on any step or have additional questions?

Here are some related questions:

1. How would the total volume change if the base reduction factor was different?
2. What if the height was a different fraction of the base side length?
3. How does the volume formula change if the base is not square?
4. What is the impact of altering the common ratio in a geometric sequence?
5. How can we calculate the side length from a given volume?

Tip: In geometric sequences, understanding the common ratio’s impact helps in quickly determining patterns for sums and growth rates.