Question
An illustration shows a 3 D triangular prism. The edges that are not visible are represented by dashed lines. The length and base of the triangular prism measures 12 centimeters and 4 centimeters. The perpendicular height h measures 6 centimeters, and the sides of the triangular face measures 7 centimeters.
Jordan purchased a piece of fabric that is 18 cm by 18 cm in order to make a tent for her little sister’s dolls. Using the dimensions for the tent shaped as a triangular prism, how much fabric will be left over?
(1 point)
Responses
324 cm2
324 cm squared
240 cm2
240 cm squared
184 cm2
184 cm squared
84 cm2
Jordan purchased a piece of fabric that is 18 cm by 18 cm in order to make a tent for her little sister’s dolls. Using the dimensions for the tent shaped as a triangular prism, how much fabric will be left over?
(1 point)
Responses
324 cm2
324 cm squared
240 cm2
240 cm squared
184 cm2
184 cm squared
84 cm2
Answers
GPT-4o mini
To determine how much fabric will be left over after making the tent in the shape of a triangular prism, we need to calculate the surface area of the triangular prism and compare it to the area of the fabric Jordan purchased.
1. **Calculate the area of one triangular face:**
The formula for the area of a triangle is:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
The base of the triangular face is 4 cm, and the height \(h\) is 6 cm.
\[
A = \frac{1}{2} \times 4 \text{ cm} \times 6 \text{ cm} = \frac{1}{2} \times 24 \text{ cm}^2 = 12 \text{ cm}^2
\]
2. **Calculate the area of the three rectangular faces:**
The rectangular faces are formed from the sides of the triangular face and the length of the prism (which is 12 cm):
- The first rectangle (base) has area:
\[
\text{Area} = \text{base} \times \text{length} = 4 \text{ cm} \times 12 \text{ cm} = 48 \text{ cm}^2
\]
- The second rectangle (side measure 7 cm) has area:
\[
\text{Area} = 7 \text{ cm} \times 12 \text{ cm} = 84 \text{ cm}^2
\]
- The third rectangle (another side measure 7 cm) also has the same area:
\[
\text{Area} = 7 \text{ cm} \times 12 \text{ cm} = 84 \text{ cm}^2
\]
3. **Add the areas of the two triangular faces and three rectangular faces:**
- Total area of the two triangular faces:
\[
2 \times 12 \text{ cm}^2 = 24 \text{ cm}^2
\]
- Total area of the three rectangular faces:
\[
48 \text{ cm}^2 + 84 \text{ cm}^2 + 84 \text{ cm}^2 = 216 \text{ cm}^2
\]
- The total surface area of the triangular prism is:
\[
\text{Total Surface Area} = 24 \text{ cm}^2 + 216 \text{ cm}^2 = 240 \text{ cm}^2
\]
4. **Calculate the total area of the fabric purchased:**
The fabric purchased is 18 cm by 18 cm, thus the area is:
\[
18 \text{ cm} \times 18 \text{ cm} = 324 \text{ cm}^2
\]
5. **Calculate the leftover fabric:**
The fabric left over after making the tent is:
\[
\text{Leftover Fabric} = \text{Area of Fabric} - \text{Surface Area of Tent} = 324 \text{ cm}^2 - 240 \text{ cm}^2 = 84 \text{ cm}^2
\]
Therefore, the amount of fabric that will be left over is **84 cm²**.
1. **Calculate the area of one triangular face:**
The formula for the area of a triangle is:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
The base of the triangular face is 4 cm, and the height \(h\) is 6 cm.
\[
A = \frac{1}{2} \times 4 \text{ cm} \times 6 \text{ cm} = \frac{1}{2} \times 24 \text{ cm}^2 = 12 \text{ cm}^2
\]
2. **Calculate the area of the three rectangular faces:**
The rectangular faces are formed from the sides of the triangular face and the length of the prism (which is 12 cm):
- The first rectangle (base) has area:
\[
\text{Area} = \text{base} \times \text{length} = 4 \text{ cm} \times 12 \text{ cm} = 48 \text{ cm}^2
\]
- The second rectangle (side measure 7 cm) has area:
\[
\text{Area} = 7 \text{ cm} \times 12 \text{ cm} = 84 \text{ cm}^2
\]
- The third rectangle (another side measure 7 cm) also has the same area:
\[
\text{Area} = 7 \text{ cm} \times 12 \text{ cm} = 84 \text{ cm}^2
\]
3. **Add the areas of the two triangular faces and three rectangular faces:**
- Total area of the two triangular faces:
\[
2 \times 12 \text{ cm}^2 = 24 \text{ cm}^2
\]
- Total area of the three rectangular faces:
\[
48 \text{ cm}^2 + 84 \text{ cm}^2 + 84 \text{ cm}^2 = 216 \text{ cm}^2
\]
- The total surface area of the triangular prism is:
\[
\text{Total Surface Area} = 24 \text{ cm}^2 + 216 \text{ cm}^2 = 240 \text{ cm}^2
\]
4. **Calculate the total area of the fabric purchased:**
The fabric purchased is 18 cm by 18 cm, thus the area is:
\[
18 \text{ cm} \times 18 \text{ cm} = 324 \text{ cm}^2
\]
5. **Calculate the leftover fabric:**
The fabric left over after making the tent is:
\[
\text{Leftover Fabric} = \text{Area of Fabric} - \text{Surface Area of Tent} = 324 \text{ cm}^2 - 240 \text{ cm}^2 = 84 \text{ cm}^2
\]
Therefore, the amount of fabric that will be left over is **84 cm²**.