To solve the problem, we need to determine the surface area of the given 3D shape. The shape appears to be a combination of a rectangular prism and a cube, with some faces overlapping or shared.
Step-by-Step Solution:
####
1. Identify the Components:
The shape consists of:
- A rectangular prism with dimensions \(15 \, \text{cm} \times 5 \, \text{cm} \times 5 \, \text{cm}\).
- A cube with side length \(5 \, \text{cm}\).
The cube is attached to one of the faces of the rectangular prism, specifically the face with dimensions \(5 \, \text{cm} \times 5 \, \text{cm}\).
####
2. Calculate the Surface Area of Each Component:
#####
Rectangular Prism:
The surface area of a rectangular prism is given by:
\[
\text{Surface Area} = 2(lw + lh + wh)
\]
where \(l = 15 \, \text{cm}\), \(w = 5 \, \text{cm}\), and \(h = 5 \, \text{cm}\).
\[
\text{Surface Area of Rectangular Prism} = 2(15 \times 5 + 15 \times 5 + 5 \times 5)
\]
\[
= 2(75 + 75 + 25)
\]
\[
= 2 \times 175
\]
\[
= 350 \, \text{cm}^2
\]
#####
Cube:
The surface area of a cube is given by:
\[
\text{Surface Area} = 6s^2
\]
where \(s = 5 \, \text{cm}\).
\[
\text{Surface Area of Cube} = 6 \times 5^2
\]
\[
= 6 \times 25
\]
\[
= 150 \, \text{cm}^2
\]
####
3. Adjust for Overlapping Faces:
When the cube is attached to the rectangular prism, one face of the cube (which is \(5 \, \text{cm} \times 5 \, \text{cm}\)) overlaps with one face of the rectangular prism. This means we need to subtract the area of this overlapping face from the total surface area.
- Area of the overlapping face:
\[
5 \, \text{cm} \times 5 \, \text{cm} = 25 \, \text{cm}^2
\]
####
4. Calculate the Total Surface Area:
The total surface area is the sum of the surface areas of the rectangular prism and the cube, minus the area of the overlapping face.
\[
\text{Total Surface Area} = (\text{Surface Area of Rectangular Prism}) + (\text{Surface Area of Cube}) - (\text{Area of Overlapping Face})
\]
\[
= 350 + 150 - 25
\]
\[
= 475 \, \text{cm}^2
\]
Final Answer:
\[
\boxed{475}
\]
Parent Tip: Review the logic above to help your child master the concept of composite figure.
