The image shows a geometric figure consisting of two pyramids with square bases, each with a side length of 24 cm. The upper pyramid has a height of 16 cm, and the lower pyramid has a height of 9 cm. The task is to calculate the surface area of this composite figure.

Steps to Solve:

1. Base Area: The base of both pyramids is a square with side length $24 \, \text{cm}$. $\text{Base Area} = 24 \times 24 = 576 \, \text{cm}^2$

2. Slant Height Calculation: To find the slant height of the upper and lower pyramids, we use the Pythagorean theorem. The slant height is the hypotenuse formed by half of the base side (12 cm) and the pyramid height.

   • For the upper pyramid: $\text{Slant Height} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \, \text{cm}$

   • For the lower pyramid: $\text{Slant Height} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \, \text{cm}$

3. Lateral Surface Area: The lateral surface area of each pyramid is composed of four triangular faces. The area of one triangular face is: $\text{Area of one triangle} = \frac{1}{2} \times \text{base side} \times \text{slant height}$

   • For the upper pyramid: $\text{Lateral Area} = 4 \times \frac{1}{2} \times 24 \times 20 = 4 \times 240 = 960 \, \text{cm}^2$

   • For the lower pyramid: $\text{Lateral Area} = 4 \times \frac{1}{2} \times 24 \times 15 = 4 \times 180 = 720 \, \text{cm}^2$

4. Total Surface Area: The total surface area is the sum of the lateral surface areas of both pyramids and the area of the base (which is shared by both pyramids). $\text{Total Surface Area} = \text{Base Area} + \text{Upper Lateral Area} + \text{Lower Lateral Area}$ $\text{Total Surface Area} = 576 + 960 + 720 = 2256 \, \text{cm}^2$

Thus, the total surface area of the figure is 2256 cm², which corresponds to option D.

Would you like a detailed explanation of any part of this solution or have any further questions?

Follow-up Questions:

1. How would the surface area change if the side length of the base were doubled?
2. What is the volume of each pyramid in this composite shape?
3. How can you find the surface area if the height of the lower pyramid changes?
4. How does the lateral surface area relate to the slant height in a pyramid?
5. How would the surface area be affected if the pyramids were made from different materials with different surface textures?

Tip:

When calculating slant height in pyramids, always remember to apply the Pythagorean theorem using half of the base's side length as one leg of the triangle.