Question
a wedge of cheese, the shape like a triangular prism how much plastic wrap is needed to cover the wedge of cheese you have you have a 12 cm, a 5 cm, a 9 cm and a 13 cm
Answers
Answered by
GPT-4o mini
To determine how much plastic wrap is needed to cover a wedge of cheese shaped like a triangular prism, we'll first need to calculate the surface area of the wedge.
A triangular prism has two triangular bases and three rectangular lateral faces.
### Given:
- We have four dimensions: 12 cm, 5 cm, 9 cm, and 13 cm. We need to clarify which dimensions represent the triangle's sides. Assuming we have a triangular base formed by dimensions 5 cm, 12 cm, and 13 cm, and the prism's height as 9 cm.
### Step 1: Verify if the given dimensions can form a triangle (5, 12, 13).
The dimensions follow the Pythagorean theorem:
\[
5^2 + 12^2 = 25 + 144 = 169 = 13^2
\]
Thus, they can form a right triangle.
### Step 2: Calculate the area of the triangular base.
The area \( A \) of a right triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, we can use 5 cm as the base and 12 cm as the height:
\[
A = \frac{1}{2} \times 5 \times 12 = 30 \text{ cm}^2
\]
### Step 3: Calculate the lateral surface area of the prism.
The height of the prism is given as 9 cm, and we need to find the areas of the three rectangular faces:
1. Rectangle with one side (5 cm):
\[
A_1 = 5 \times 9 = 45 \text{ cm}^2
\]
2. Rectangle with one side (12 cm):
\[
A_2 = 12 \times 9 = 108 \text{ cm}^2
\]
3. Rectangle with one side (13 cm):
\[
A_3 = 13 \times 9 = 117 \text{ cm}^2
\]
### Step 4: Sum the areas.
Now, we can calculate the total surface area \( SA \) of the prism:
- Surface area of two triangular bases:
\[
\text{Total Area of both bases} = 2 \times 30 = 60 \text{ cm}^2
\]
- Total lateral area:
\[
\text{Total Lateral Area} = A_1 + A_2 + A_3 = 45 + 108 + 117 = 270 \text{ cm}^2
\]
- Therefore, the total surface area \( SA \) is:
\[
SA = \text{Total Area of both bases} + \text{Total Lateral Area} = 60 + 270 = 330 \text{ cm}^2
\]
### Wrap It Up
The amount of plastic wrap needed to cover the wedge of cheese is:
\[
\boxed{330 \text{ cm}^2}
\]
A triangular prism has two triangular bases and three rectangular lateral faces.
### Given:
- We have four dimensions: 12 cm, 5 cm, 9 cm, and 13 cm. We need to clarify which dimensions represent the triangle's sides. Assuming we have a triangular base formed by dimensions 5 cm, 12 cm, and 13 cm, and the prism's height as 9 cm.
### Step 1: Verify if the given dimensions can form a triangle (5, 12, 13).
The dimensions follow the Pythagorean theorem:
\[
5^2 + 12^2 = 25 + 144 = 169 = 13^2
\]
Thus, they can form a right triangle.
### Step 2: Calculate the area of the triangular base.
The area \( A \) of a right triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, we can use 5 cm as the base and 12 cm as the height:
\[
A = \frac{1}{2} \times 5 \times 12 = 30 \text{ cm}^2
\]
### Step 3: Calculate the lateral surface area of the prism.
The height of the prism is given as 9 cm, and we need to find the areas of the three rectangular faces:
1. Rectangle with one side (5 cm):
\[
A_1 = 5 \times 9 = 45 \text{ cm}^2
\]
2. Rectangle with one side (12 cm):
\[
A_2 = 12 \times 9 = 108 \text{ cm}^2
\]
3. Rectangle with one side (13 cm):
\[
A_3 = 13 \times 9 = 117 \text{ cm}^2
\]
### Step 4: Sum the areas.
Now, we can calculate the total surface area \( SA \) of the prism:
- Surface area of two triangular bases:
\[
\text{Total Area of both bases} = 2 \times 30 = 60 \text{ cm}^2
\]
- Total lateral area:
\[
\text{Total Lateral Area} = A_1 + A_2 + A_3 = 45 + 108 + 117 = 270 \text{ cm}^2
\]
- Therefore, the total surface area \( SA \) is:
\[
SA = \text{Total Area of both bases} + \text{Total Lateral Area} = 60 + 270 = 330 \text{ cm}^2
\]
### Wrap It Up
The amount of plastic wrap needed to cover the wedge of cheese is:
\[
\boxed{330 \text{ cm}^2}
\]