https://file.aiquickdraw.com/mathbot/file/1729353866412d3md91rs.jpg

The problem in the image is about calculating the lateral surface area of a rectangular parallelepiped (прямокутний паралелепіпед) with the given dimensions:

- The sides of the base are 2 cm and 6 cm.
- The diagonal of the parallelepiped is 7 cm.

To find the lateral surface area, let's break it down:

### Step 1: Understanding the Diagonal
The diagonal \( d \) of a rectangular parallelepiped relates to its dimensions (length \( l \), width \( w \), and height \( h \)) by the formula:
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
Here, \( l = 6 \) cm, \( w = 2 \) cm, and \( d = 7 \) cm. Plugging in the known values:
\[
7 = \sqrt{6^2 + 2^2 + h^2}
\]
\[
7 = \sqrt{36 + 4 + h^2}
\]
\[
7 = \sqrt{40 + h^2}
\]
Squaring both sides:
\[
49 = 40 + h^2
\]
\[
h^2 = 9
\]
\[
h = 3 \, \text{cm}
\]

### Step 2: Lateral Surface Area Formula
The lateral surface area \( A_{\text{lateral}} \) of a rectangular parallelepiped is given by:
\[
A_{\text{lateral}} = 2h(l + w)
\]
Substitute the known values \( l = 6 \) cm, \( w = 2 \) cm, and \( h = 3 \) cm:
\[
A_{\text{lateral}} = 2 \times 3 \times (6 + 2) = 6 \times 8 = 48 \, \text{cm}^2
\]

### Final Answer:
The lateral surface area is \( 48 \, \text{cm}^2 \). 

The correct option is **48**.

Would you like more details or further explanations?

Here are some related questions you can explore:
1. What is the total surface area of the parallelepiped?
2. How would the lateral surface area change if the diagonal was longer?
3. What if the base had different side lengths, how would the height change?
4. How does the formula for diagonal change in three-dimensional figures?
5. Can you find the volume of the parallelepiped with these dimensions?

**Tip:** When dealing with 3D geometry, always visualize or sketch the object to better understand the relationships between dimensions!

Сторони основи прямокутного паралелепіпеда 2 см і 6 см, діагональ паралелепіпеда 7 см. Знайти площу бічної поверхні паралелепіпеда.

Calculate the lateral surface area of a rectangular parallelepiped with base dimensions 2 cm and 6 cm, and a diagonal of 7 cm. Learn to use the Pythagorean theorem in 3D geometry to determine the height and surface area.

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