Question

An illustration shows a three dimensional composite figure formed by a smaller rectangular prism connected to a larger rectangular prism. The top, right, and front faces are visible. The faces and edges that are not visible are indicated by dashed lines. The rectangular prism on the left has a length of 10 millimeters a height of 8 millimeters. The width of the rectangular prism is not shown, but it aligns perfectly with the width of the rectangular prism on the right. The rectangular prism on the right has a length of 6 millimeters, a width of 6 millimeters, and a height 6 of millimeters. The combined length of the two rectangular prisms measures 16 millimeters.


What is the total surface area of the figure?

(1 point)

Answer 1

To find the total surface area of the composite figure formed by the two rectangular prisms, we first need to calculate the surface area of each prism separately and then add them together, making sure to account for the area where they are connected (which will not be part of the total surface area).

### Dimensions of the Prisms:
1. **Left Rectangular Prism**:
- Length = 10 mm
- Height = 8 mm
- Width = \( w \) mm (unknown, aligned with the right prism)

2. **Right Rectangular Prism**:
- Length = 6 mm
- Width = 6 mm
- Height = 6 mm

### Step 1: Find the Width of the Left Prism
The combined length of both prisms is given as 16 mm. Therefore, the width of the left prism can be found as follows:

\[
10 \text{ mm} + 6 \text{ mm} = 16 \text{ mm}
\]

This confirms that the left prism's length is 10 mm and the right prism's length is 6 mm.

### Step 2: Calculate the Surface Area of Each Prism

#### Surface Area of the Left Rectangular Prism:
The surface area \( S \) of a rectangular prism can be calculated using the formula:
\[
S = 2(lw + lh + wh)
\]

Plugging the values we know:
- Length (\( l \)) = 10 mm
- Height (\( h \)) = 8 mm
- Width (\( w \)) = aligned with the right prism so \( w \) remains unknown until calculated. We won't need \( w \) yet since it cancels out.

The surface area of the left prism excluding the back face where it connects to the right prism is:
\[
S_{left} = 2(10w + 10 \cdot 8 + w \cdot 8) - (w \cdot 8)
\]
\[
= 2(10w + 80 + 8w) - 8w
\]
\[
= 2(18w + 80) - 8w
\]
\[
= 36w + 160 - 8w
\]
\[
= 28w + 160
\]

#### Surface Area of the Right Rectangular Prism:
For the right prism:
- Length = 6 mm
- Width = 6 mm
- Height = 6 mm

Calculating the surface area:
\[
S_{right} = 2(6 \cdot 6 + 6 \cdot 6 + 6 \cdot 6) - \text{(connection face area)}
\]
\[
= 2(36 + 36 + 36) - 36 \quad \text{(only one face connects with left prism)}
\]
\[
= 2(108) - 36
\]
\[
= 216 - 36
\]
\[
= 180 \text{ mm}^2
\]

### Step 3: Total Surface Area Calculation
Now, we add the surface areas of both prisms together:
\[
S_{total} = S_{left} + S_{right}
\]
\[
= (28w + 160) + 180
\]
\[
= 28w + 340
\]

Since we don't have the width \( w \) directly from the figure or stated values, the final equation to express the total surface area is:
\[
\text{Total Surface Area} = 28w + 340 \text{ mm}^2
\]

To fully quantify the total surface area, we need the width of the left rectangular prism, but based on the information in the question we cannot determine it since it's not provided. Therefore, the total surface area depends linearly on \( w \).

For a final numerical value, if we were to assume \( w = 6 \) (matching the right prism width), we could substitute \( w \):

\[
S_{total} = 28(6) + 340 = 168 + 340 = 508 \text{ mm}^2
\]

Thus, if \( w \) is confirmed, insert that for the final surface area. For now, it's expressed in terms of \( w \):
\[
\text{Total Surface Area} = 28w + 340 \text{ mm}^2
\]

Without width given, we cannot finalize the numeric value.