To find the height of the original pyramid from which the frustum was obtained, we need to understand the relationship between the dimensions of the frustum and the original pyramid. Let's break this down step-by-step. ### Step 1: Understanding the Frustum of a Pyramid A frustum of a pyramid is formed when the top part of the pyramid is cut off by a plane parallel to the base. In this case, we have: - The lower base (larger square) with a side length of 5 m. - The upper base (smaller square) with a side length of 2 m. - The vertical distance (height of the frustum) between the two bases is 6 m. ### Step 2: Visualizing the Problem Imagine a pyramid with a square base. The larger base is at the bottom, and the smaller base is at the top. The height of the frustum is the distance between these two bases. ### Step 3: Using Similar Triangles When the top of the pyramid is sliced off, the remaining frustum and the original pyramid share similar triangles. The height of the original pyramid can be found using the properties of similar triangles. Let: - \( H \) = height of the original pyramid - \( h \) = height of the frustum = 6 m - \( b_1 \) = side length of the lower base = 5 m - \( b_2 \) = side length of the upper base = 2 m ### Step 4: Setting Up the Proportions The height of the original pyramid can be expressed in terms of the height of the frustum and the dimensions of the bases. The relationship can be set up as follows: 1. The total height of the original pyramid can be divided into two parts: the height of the frustum (6 m) and the height of the smaller pyramid that was removed (let's call this \( h_1 \)). 2. The ratio of the heights of the original pyramid and the smaller pyramid is equal to the ratio of the bases: \[ \frac{H}{h_1} = \frac{b_1}{b_2} \] Substituting the values we have: \[ \frac{H}{h_1} = \frac{5}{2} \] ### Step 5: Expressing \( h_1 \) in Terms of \( H \) From the above proportion, we can express \( h_1 \): \[ h_1 = \frac{2}{5} H \] ### Step 6: Relating \( H \) and \( h_1 \) to the Height of the Frustum The total height of the original pyramid \( H \) can be expressed as: \[ H = h_1 + h \] Substituting \( h_1 \): \[ H = \frac{2}{5} H + 6 \] ### Step 7: Solving for \( H \) Now, we can solve for \( H \): 1. Multiply through by 5 to eliminate the fraction: \[ 5H = 2H + 30 \] 2. Rearranging gives: \[ 5H - 2H = 30 \] \[ 3H = 30 \] 3. Dividing both sides by 3: \[ H = 10 \] ### Conclusion The height of the original pyramid is **10 m**. ### Step 8: Analyzing the Options - **Option A (8.0 m)**: Incorrect. This does not satisfy the relationship derived from the similar triangles. - **Option B (8.4 m)**: Incorrect. This value does not match the calculated height. - **Option C (9.0 m)**: Incorrect. This is also not the correct height based on our calculations. - **Option D (10.0 m)**: Correct. This matches our derived height of the original pyramid. ### Revision Summary - A frustum of a pyramid is formed by slicing off the top of the pyramid. - The height of the original pyramid can be found using the properties of similar triangles. - The relationship between the heights and the bases leads to a proportion that can be solved for the height. - The final height of the original pyramid is 10 m, confirming that option D is correct.