The shape in the image represents a frustum of a cone (truncated cone) with a cylindrical portion on top, often used to model swimming pools. To find the surface area of this shape, we will consider both the lateral surface areas of the frustum and the cylindrical section, as well as the areas of the top and bottom circles. Here are the steps:

1. Lateral Surface Area of the Cylindrical Section:

The lateral surface area of a cylinder is given by: $A_{\text{cylinder}} = 2\pi L_1 D_S$ where:

• $L_1$ is the surface radius (top radius),
• $D_S$ is the shallow end depth (the height of the cylindrical portion).

2. Lateral Surface Area of the Frustum:

The formula for the lateral surface area of a frustum is: $A_{\text{frustum}} = \pi (L_1 + L_2) \times \sqrt{(L_1 - L_2)^2 + D_D^2}$ where:

• $L_1$ is the surface radius (top radius),
• $L_2$ is the bottom radius,
• $D_D$ is the deep end depth (the height of the frustum part).

3. Area of the Top Circle (at the surface):

The area of the top circle is: $A_{\text{top}} = \pi L_1^2$

4. Area of the Bottom Circle (at the bottom of the frustum):

The area of the bottom circle is: $A_{\text{bottom}} = \pi L_2^2$

5. Total Surface Area:

The total surface area is the sum of all the individual areas: $A_{\text{total}} = A_{\text{cylinder}} + A_{\text{frustum}} + A_{\text{top}} + A_{\text{bottom}}$ Substitute the corresponding values into the equations to compute the total surface area.

Do you want me to proceed with the calculations if you provide the values for $L_1$, $L_2$, $D_S$, and $D_D$, or do you need more clarifications?

Related Questions:

1. How do you find the volume of this frustum and cylinder combination?
2. How would the surface area change if the pool were not truncated (i.e., if $L_2 = 0$)?
3. What is the significance of the lateral surface area in real-world applications like pool design?
4. How do you calculate the surface area of a general frustum with any given slope angle?
5. How do you compare the surface area of this shape with a simple cylindrical tank of similar height?

Tip:

For accurate surface area calculations, always check the consistency of the dimensions provided, especially when dealing with composite shapes like this.