To find out how much fabric will be left over, we first need to calculate the surface area of the triangular prism that Jordan wants to make. The tent consists of two triangular bases and three rectangular sides.
Step 1: Calculate the area of the triangular base.
The base of the triangular prism has a base length of 4 cm and a height (perpendicular) of 6 cm. The area \( A \) of the triangular base can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Inserting the values we know:
\[ A = \frac{1}{2} \times 4 , \text{cm} \times 6 , \text{cm} = 12 , \text{cm}^2 \]
Since there are two triangular bases, the total area of the triangular bases is:
\[ \text{Total area of triangular bases} = 2 \times 12 , \text{cm}^2 = 24 , \text{cm}^2 \]
Step 2: Calculate the area of the rectangular sides.
The prism has three rectangular sides. Their dimensions are:
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Two rectangles with dimensions \( 12 , \text{cm} \times 4 , \text{cm} \): \[ \text{Area} = 12 , \text{cm} \times 4 , \text{cm} = 48 , \text{cm}^2 , \text{(each)} \] Since there are 2 such rectangles: \[ \text{Total area for 2 rectangles} = 2 \times 48 , \text{cm}^2 = 96 , \text{cm}^2 \]
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One rectangle with dimensions \( 12 , \text{cm} \times 7 , \text{cm} \): \[ \text{Area} = 12 , \text{cm} \times 7 , \text{cm} = 84 , \text{cm}^2 \]
Step 3: Sum the areas of the rectangular sides.
The total area for all three rectangular sides is:
\[ \text{Total area of rectangular sides} = 96 , \text{cm}^2 + 84 , \text{cm}^2 = 180 , \text{cm}^2 \]
Step 4: Calculate the total surface area of the prism.
The total surface area \( SA \) of the prism is:
\[ SA = \text{Area of triangular bases} + \text{Area of rectangular sides} \] \[ SA = 24 , \text{cm}^2 + 180 , \text{cm}^2 = 204 , \text{cm}^2 \]
Step 5: Calculate the area of the fabric that Jordan purchased.
Jordan purchased fabric measuring \( 18 , \text{cm} \times 18 , \text{cm} \):
\[ \text{Area of fabric} = 18 , \text{cm} \times 18 , \text{cm} = 324 , \text{cm}^2 \]
Step 6: Find the leftover fabric.
To find the amount of fabric left over, we subtract the surface area of the prism from the area of the fabric:
\[ \text{Leftover fabric} = \text{Area of fabric} - \text{Total surface area} \] \[ \text{Leftover fabric} = 324 , \text{cm}^2 - 204 , \text{cm}^2 = 120 , \text{cm}^2 \]
However, this value does not match the answer options provided. Let's review the calculations to ensure everything has been accounted accurately.
- Surface area of triangular bases: \( 24 , \text{cm}^2 \)
- Surface area of rectangular sides: \( 180 , \text{cm}^2 \)
So total should be:
\[ 204 , \text{cm}^2 \]
Leftover fabric:
\[ 324 , \text{cm}^2 - 204 , \text{cm}^2 = 120 , \text{cm}^2 \]
None of the given options matches the calculated leftover fabric, leading to a potential misunderstanding or miscommunication in the problem prompt.
Could you please verify whether the dimensions provided are accurate? Based on the correct options available, I would suggest rounding or reconsidering the triangle dimensions used. Reassess for the fabric initially purchased, as it leads to different conclusions based on provided options which are:
- 240 cm²
- 84 cm²
- 324 cm²
- 184 cm²
After reconciling everything, though, logically no significant options correlate—readjust.
Assuming reassessment reveals overlooked errors initially, if correcting leads to one of those figures based on new inputs, explore queries again if needed on specific recalibrations needed based on triangular perimeters assessed or heights observed.