Question

Leon has a piece of cheese shaped like a triangular prism that he needs to wrap in plastic wrap.

A triangular prism. The triangular base has a base of 4 inches and height of 3 inches. The height of the prism is 7 inches.


What is the minimum amount of plastic wrap that he needs to cover the entire piece of cheese?
42 inches squared
84 inches squared
96 inches squared
108 inches squared

Answer 1

To find the minimum amount of plastic wrap that Leon needs to cover the triangular prism, we first need to calculate the surface area of the prism.

A triangular prism consists of two triangular bases and three rectangular lateral faces.

### Step 1: Calculate the area of the triangular base

The area \(A\) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

For the triangular base:
- Base = 4 inches
- Height = 3 inches

Calculating the area:
\[
A = \frac{1}{2} \times 4 \times 3 = \frac{12}{2} = 6 \, \text{inches}^2
\]

### Step 2: Calculate the areas of the rectangular faces

The height of the prism is given as 7 inches. The rectangular faces can be calculated as follows:

1. **Rectangular face corresponding to the triangle's base (4 inches)**:
\[
\text{Area}_{base} = \text{base} \times \text{height} = 4 \times 7 = 28 \, \text{inches}^2
\]

2. **Rectangular face corresponding to the triangle's height (3 inches)**:
\[
\text{Area}_{height} = \text{height} \times \text{height} = 3 \times 7 = 21 \, \text{inches}^2
\]

3. **Rectangular face corresponding to the slant height (hypotenuse)**:
To find the length of the hypotenuse, we use the Pythagorean theorem. The hypotenuse \(c\) of the triangle is:
\[
c = \sqrt{(\text{base})^2 + (\text{height})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{inches}
\]
The area of this face is:
\[
\text{Area}_{hypotenuse} = \text{hypotenuse} \times \text{height} = 5 \times 7 = 35 \, \text{inches}^2
\]

### Step 3: Calculate the total surface area

The total surface area \(SA\) of the prism is the sum of the areas of the two triangular bases and the three rectangular faces:
\[
SA = 2 \times \text{Area}_{triangle} + \text{Area}_{base} + \text{Area}_{height} + \text{Area}_{hypotenuse}
\]
Plugging in the values:
\[
SA = 2 \times 6 + 28 + 21 + 35
\]
\[
SA = 12 + 28 + 21 + 35 = 96 \, \text{inches}^2
\]

### Conclusion

The minimum amount of plastic wrap that Leon needs is 96 inches squared. The answer is:

**96 inches squared**