Question
Jordan purchased a piece of fabric that is 18 cm x 18 cm in order to make a tent for her little sister’s dolls, using the dimensions of the tent shaped as a triangular prism how much fabric will be left over
in the diagram you have the triangular base is 4 cm the height 7 cm the rectangular base is 12 cm x 6 cm
in the diagram you have the triangular base is 4 cm the height 7 cm the rectangular base is 12 cm x 6 cm
Answers
GPT-4o mini
To determine how much fabric Jordan will have left over after making the tent, we first need to calculate the area of the triangular prism shaped tent and then subtract that from the total area of the fabric she purchased.
### Step 1: Calculate the area of the fabric purchased
Jordan has a square piece of fabric that is 18 cm x 18 cm:
\[
\text{Area of the fabric} = 18 \, \text{cm} \times 18 \, \text{cm} = 324 \, \text{cm}^2
\]
### Step 2: Calculate the area of the tent
The tent has a triangular base and rectangular sides. We will first calculate the areas of these components.
#### Area of the triangular base
The triangular base has a base (b) of 4 cm and a height (h) of 7 cm. The area of a triangle is given by the formula:
\[
\text{Area of triangle} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 4 \, \text{cm} \times 7 \, \text{cm} = \frac{1}{2} \times 28 \, \text{cm}^2 = 14 \, \text{cm}^2
\]
Since this triangular base is at both the top and bottom of the prism, we will have two of these triangles:
\[
\text{Total triangular area} = 2 \times 14 \, \text{cm}^2 = 28 \, \text{cm}^2
\]
#### Area of the rectangular sides
The prism has three rectangular sides:
1. Front and back rectangles: Each has dimensions of 4 cm (width) and 12 cm (length).
\[
\text{Area of each rectangular face} = 4 \, \text{cm} \times 12 \, \text{cm} = 48 \, \text{cm}^2
\]
So, for both front and back:
\[
\text{Total area for front and back} = 2 \times 48 \, \text{cm}^2 = 96 \, \text{cm}^2
\]
2. Two side rectangles: Each has dimensions of 6 cm (width) and 12 cm (length).
\[
\text{Area of each side face} = 6 \, \text{cm} \times 12 \, \text{cm} = 72 \, \text{cm}^2
\]
So, for both side rectangles:
\[
\text{Total area for both sides} = 2 \times 72 \, \text{cm}^2 = 144 \, \text{cm}^2
\]
#### Total area of the tent
Now we sum all of the areas:
\[
\text{Total area of the tent} = \text{Total triangular area} + \text{Total area for front and back} + \text{Total area for both sides}
\]
\[
= 28 \, \text{cm}^2 + 96 \, \text{cm}^2 + 144 \, \text{cm}^2 = 268 \, \text{cm}^2
\]
### Step 3: Calculate the leftover fabric
Now we subtract the area of the tent from the area of the fabric:
\[
\text{Leftover fabric} = \text{Area of fabric} - \text{Total area of the tent}
= 324 \, \text{cm}^2 - 268 \, \text{cm}^2 = 56 \, \text{cm}^2
\]
### Conclusion
Jordan will have **56 cm²** of fabric leftover after making the tent.
### Step 1: Calculate the area of the fabric purchased
Jordan has a square piece of fabric that is 18 cm x 18 cm:
\[
\text{Area of the fabric} = 18 \, \text{cm} \times 18 \, \text{cm} = 324 \, \text{cm}^2
\]
### Step 2: Calculate the area of the tent
The tent has a triangular base and rectangular sides. We will first calculate the areas of these components.
#### Area of the triangular base
The triangular base has a base (b) of 4 cm and a height (h) of 7 cm. The area of a triangle is given by the formula:
\[
\text{Area of triangle} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 4 \, \text{cm} \times 7 \, \text{cm} = \frac{1}{2} \times 28 \, \text{cm}^2 = 14 \, \text{cm}^2
\]
Since this triangular base is at both the top and bottom of the prism, we will have two of these triangles:
\[
\text{Total triangular area} = 2 \times 14 \, \text{cm}^2 = 28 \, \text{cm}^2
\]
#### Area of the rectangular sides
The prism has three rectangular sides:
1. Front and back rectangles: Each has dimensions of 4 cm (width) and 12 cm (length).
\[
\text{Area of each rectangular face} = 4 \, \text{cm} \times 12 \, \text{cm} = 48 \, \text{cm}^2
\]
So, for both front and back:
\[
\text{Total area for front and back} = 2 \times 48 \, \text{cm}^2 = 96 \, \text{cm}^2
\]
2. Two side rectangles: Each has dimensions of 6 cm (width) and 12 cm (length).
\[
\text{Area of each side face} = 6 \, \text{cm} \times 12 \, \text{cm} = 72 \, \text{cm}^2
\]
So, for both side rectangles:
\[
\text{Total area for both sides} = 2 \times 72 \, \text{cm}^2 = 144 \, \text{cm}^2
\]
#### Total area of the tent
Now we sum all of the areas:
\[
\text{Total area of the tent} = \text{Total triangular area} + \text{Total area for front and back} + \text{Total area for both sides}
\]
\[
= 28 \, \text{cm}^2 + 96 \, \text{cm}^2 + 144 \, \text{cm}^2 = 268 \, \text{cm}^2
\]
### Step 3: Calculate the leftover fabric
Now we subtract the area of the tent from the area of the fabric:
\[
\text{Leftover fabric} = \text{Area of fabric} - \text{Total area of the tent}
= 324 \, \text{cm}^2 - 268 \, \text{cm}^2 = 56 \, \text{cm}^2
\]
### Conclusion
Jordan will have **56 cm²** of fabric leftover after making the tent.