d. 264 \({ cm} ^ { 2} \)

UpStudy Free Solution:

To find the surface area of the triangular prism with the corrected dimensions of the three rectangular faces, let's calculate the areas step by step again.

Step 1: Calculate the area of the triangular base

The base of the triangular prism is a right triangle with legs of 6 cm and 8 cm.

The area \(A\) of a right triangle is given by:

\(A = \frac { 1} { 2} \times \text { base} \times \text { height} \)

For our triangle:

\(A = \frac { 1} { 2} \times 6 \space \text { cm} \times 8 \space \text { cm} = \frac { 1} { 2} \times 48 \space \text { cm} ^ 2 = 24 \space \text { cm} ^ 2\)

Since there are two triangular bases:

\(\text { Total area of triangular bases} = 2 \times 24 \space \text { cm} ^ 2 = 48 \space \text { cm} ^ 2\)

Step 2: Calculate the area of the three rectangular faces

The three rectangular faces have the following dimensions:

1. \(6 \space \text { cm} \times 9 \space \text { cm} \)

2. \(8 \space \text { cm} \times 9 \space \text { cm} \)

3. \(10 \space \text { cm} \times 9 \space \text { cm} \)

The areas are calculated as follows:

1.

\(\text { Area} = 6 \space \text { cm} \times 9 \space \text { cm} = 54 \space \text { cm} ^ 2\)

2.

\(\text { Area} = 8 \space \text { cm} \times 9 \space \text { cm} = 72 \space \text { cm} ^ 2\)

3.

\(\text { Area} = 10 \space \text { cm} \times 9 \space \text { cm} = 90 \space \text { cm} ^ 2\)

Step 3: Sum up the areas

Adding the areas of the three rectangular faces:

\(\text { Total area of rectangular faces} = 54 \space \text { cm} ^ 2 + 72 \space \text { cm} ^ 2 + 90 \space \text { cm} ^ 2 = 216 \space \text { cm} ^ 2\)

Adding the areas of the triangular bases and the rectangular faces gives the total surface area:

\(\text { Total surface area} = 48 \space \text { cm} ^ 2 + 216 \space \text { cm} ^ 2 = 264 \space \text { cm} ^ 2\)

Supplemental Knowledge:

When calculating the surface area of a triangular prism, it is essential to account for both the areas of the triangular bases and the rectangular faces. The total surface area is the sum of the areas of all these faces.