To determine how much fabric Jordan will have 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 then subtract that from the total area of the fabric she has.
Step 1: Calculate the Surface Area of the Triangular Prism
The surface area \( SA \) of a triangular prism can be calculated using the following formula: \[ SA = B + L \] where:
- \( B \) is the area of the two triangular bases,
- \( L \) is the area of the rectangular lateral faces.
1.1 Area of the Triangular Base
The triangular base is made up of a base \( b \), height \( h \), and two sides. From the information given:
- Base \( b = 4 \) cm,
- Height \( h = 6 \) cm,
- The sides are \( 7 \) cm.
The area \( A \) of the triangular base can be calculated using the formula for the area of a triangle: \[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 6 = 12 \text{ cm}^2 \]
Since there are two triangular bases: \[ B = 2 \times A = 2 \times 12 = 24 \text{ cm}^2 \]
1.2 Area of the Rectangular Lateral Faces
The lateral faces consist of three rectangles:
- One face that corresponds to the rectangular side with dimensions length and base = \( 12 \) cm (length of prism) and \( 4 \) cm (base of triangle).
- Two faces corresponding to the altitude of the triangular face, which has dimensions of height (perpendicular height \( h = 6 \) cm) multiplied by the length of the sides of the triangle.
Calculating the area of each rectangle:
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For the rectangle corresponding to the base (which is the same as the prism length): \[ A_1 = \text{length} \times \text{base} = 12 \times 4 = 48 \text{ cm}^2 \]
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For the two rectangles corresponding to the triangular sides (each has height 6 cm): \[ A_2 = \text{length} \times \text{height} = 12 \times 7 = 84 \text{ cm}^2 \text{ (for one side)} \] Therefore, for two sides: \[ 2 \times 84 = 168 \text{ cm}^2 \]
Thus, the total lateral area \( L \) is: \[ L = A_1 + 2 \times A_2 = 48 + 168 = 216 \text{ cm}^2 \]
1.3 Total Surface Area
Finally, we can calculate the total surface area \( SA \): \[ SA = B + L = 24 + 216 = 240 \text{ cm}^2 \]
Step 2: Total Area of the Fabric
The fabric Jordan purchased is a square that measures \( 18 , \text{cm} \times 18 , \text{cm} \): \[ \text{Total area of the fabric} = 18 \times 18 = 324 \text{ cm}^2 \]
Step 3: Calculate Leftover Fabric
Now, to find how much fabric will be left over after creating the tent: \[ \text{Leftover fabric} = \text{Total area of the fabric} - \text{Surface area of the tent} = 324 - 240 = 84 \text{ cm}^2 \]
Conclusion
The amount of fabric that will be left over is \( \boxed{84 , \text{cm}^2} \).