Question

Question content area top
Part 1
Find the surface area of the triangular prism. The base of the prism is an isosceles triangle.
35 cm
37 cm
nbsp 47 cm
24 cm
Question content area bottom
Part 1
The surface area is
  
enter your response here cm Superscript 2.

Answer 1

To find the surface area of a triangular prism, we will use the formula:

\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]

### 1. Calculate the Area of the Triangular Base

The base is an isosceles triangle. For an isosceles triangle, the area can be calculated with:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

In this case, it appears that:
- The sides of the isosceles triangle are 35 cm and 37 cm (assuming 37 cm is the length of the equal sides and 35 cm is the base).
- The height of the triangle is determined using the Pythagorean theorem.

First, we need to find the height of the triangle. To find the height \( h \) from the top vertex to the base (which splits the base into two equal halves), we do the following calculations:

Let half the base be \( \frac{35}{2} = 17.5 \, \text{cm} \).
Now, we use the Pythagorean theorem:

\[
h^2 + (17.5)^2 = (37)^2
\]

Calculating \( 37^2 \) and \( (17.5)^2 \):

\[
37^2 = 1369
\]
\[
(17.5)^2 = 306.25
\]

Now substituting back into the equation:

\[
h^2 + 306.25 = 1369
\]
\[
h^2 = 1369 - 306.25 = 1062.75
\]
\[
h = \sqrt{1062.75} \approx 32.6 \, \text{cm}
\]

Now substitute back to find the area of the triangle:

\[
\text{Area} = \frac{1}{2} \times 35 \, \text{cm} \times 32.6 \, \text{cm} \approx 570.5 \, \text{cm}^2
\]

### 2. Calculate the Lateral Area

The lateral area is calculated as the perimeter of the base triangle times the height of the prism:

- The perimeter \( P \) of the triangular base is:

\[
P = 35 + 37 + 37 = 109 \, \text{cm}
\]

- If the height (length) of the prism is 24 cm, then:

\[
\text{Lateral Area} = P \times \text{height} = 109 \, \text{cm} \times 24 \, \text{cm} = 2616 \, \text{cm}^2
\]

### 3. Calculate the Total Surface Area

Now sum the areas:

\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 570.5 \, \text{cm}^2 + 2616 \, \text{cm}^2 \approx 3186.5 \, \text{cm}^2
\]

### Conclusion

The total surface area of the triangular prism is approximately:

\[
\text{Surface Area} \approx 3186.5 \, \text{cm}^2
\]

You may want to round the answer to one decimal place or whole number depending on the requirement.

Thus, you can enter this value in your response box.