To determine how much fabric will be left over, we need to calculate the total surface area of the triangular prism and then compare it to the area of the fabric Jordan purchased.
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Calculate the area of the triangular base: The triangular base has a base of 4 cm, a height (perpendicular height) of 6 cm, and two equal sides of 7 cm.
To find the area of the triangle: \[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 6 = 12 , \text{cm}^2 \]
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Calculate the surface area of the triangular prism: The surface area (SA) of a triangular prism can be calculated using the formula: \[ SA = 2 \times \text{Area of base} + \text{Perimeter of base} \times \text{length} \] where:
- The perimeter of the triangular base can be calculated as: \[ \text{Perimeter} = 4 + 7 + 7 = 18 , \text{cm} \]
- The length of the prism is given as 12 cm.
Now, substituting the values into the surface area formula: \[ SA = 2 \times 12 + 18 \times 12 \] \[ SA = 24 + 216 = 240 , \text{cm}^2 \]
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Calculate the area of the fabric: The fabric is a square piece measuring 18 cm by 18 cm: \[ \text{Area of fabric} = 18 \times 18 = 324 , \text{cm}^2 \]
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Calculate the leftover fabric: Now, subtract the surface area of the tent from the area of the fabric: \[ \text{Leftover fabric} = \text{Area of fabric} - \text{SA of prism} \] \[ \text{Leftover fabric} = 324 - 240 = 84 , \text{cm}^2 \]
Therefore, the amount of fabric that will be left over is 84 cm².