Question

Use the image to answer the question.

An illustration shows a 3 D triangular prism with the triangular face as a right triangle. The edges that are not visible are represented by dashed lines. The length and base of the triangular prism measures 30.5 centimeters and 27.9 centimeters. The perpendicular side of the triangular face measures 5.1 centimeter and the hypotenuse measures 28.4 centimeter.

A toy came in a container that is shaped like a triangular prism. How much wrapping paper is needed to wrap the toy? Round your answer to the nearest whole number.

(1 point)
S.A.=
cm2
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Answer 1

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

\[
S.A. = \text{Base Area} + \text{Lateral Area}
\]

### Step 1: Calculate the Area of the Triangular Base

The triangle given is a right triangle. The area \( A \) of a right triangle can be calculated using the formula:

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

In this case:
- Base = 27.9 cm
- Height (perpendicular side) = 5.1 cm

\[
A = \frac{1}{2} \times 27.9 \times 5.1 = \frac{1}{2} \times 142.29 \approx 71.145 \ \text{cm}^2
\]

### Step 2: Calculate the Lateral Surface Area

The lateral area consists of the three rectangles that wrap around the sides of the triangular prism. The lateral surface area \( L \) can be calculated as follows:

\[
L = \text{Perimeter of the base triangle} \times \text{length}
\]

**Step 2.1: Calculate the sides of the triangular base**

Perimeter includes:
- Base = 27.9 cm
- Height = 5.1 cm
- Hypotenuse = 28.4 cm

\[
\text{Perimeter} = 27.9 + 5.1 + 28.4 = 61.4 \ \text{cm}
\]

**Step 2.2: Calculate the lateral area**

\[
L = \text{Perimeter} \times \text{length} = 61.4 \times 30.5
\]

Calculating:

\[
L = 1873.7 \ \text{cm}^2
\]

### Step 3: Calculate Total Surface Area

Combining the area of the triangular base and the lateral area:

Since there are two triangular bases in a prism, we add the area of the triangle twice:

\[
S.A. = 2 \times A + L = 2 \times 71.145 + 1873.7
\]

Calculating:

\[
S.A. = 142.29 + 1873.7 = 2016.19 \ \text{cm}^2
\]

### Step 4: Round to the Nearest Whole Number

Rounding \( 2016.19 \) gives us:

\[
S.A. = 2016 \ \text{cm}^2
\]

### Final Answer

The amount of wrapping paper needed to wrap the toy is \( \boxed{2016} \ \text{cm}^2 \).