To find the area of the triangular base of the triangular prism using the provided information about the rectangular areas, we can follow these steps:
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Understand the net structure: The middle rectangle has an area of 64 square inches and the height of the triangular prism is represented by the 6-inch side of the right triangles attached to it.
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Find the dimensions of the rectangles:
- The left rectangle has an area of 48 square inches. Assuming it has a base \( b_1 \) and height \( h \): \[ b_1 \cdot h = 48 \quad (1) \]
- The middle rectangle has an area of 64 square inches: \[ b_2 \cdot h = 64 \quad (2) \]
- The right rectangle has an area of 80 square inches: \[ b_3 \cdot h = 80 \quad (3) \]
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Area of triangular base: The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the height is 6 inches (as given), and the base is the common base side shared with the middle rectangle.
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Establish relationships: Since the widths of these rectangles relate to the dimensions of the triangular base and height:
- From (1), (2), and (3), you can express the relationships of base lengths.
- From rectangle 2 (the middle rectangle), we know its area for base \( b \): \[ b_2 \cdot 6 = 64 \quad \Rightarrow \quad b_2 = \frac{64}{6} \approx 10.67 \text{ inches (which is \( \text{base} \)), this determines the base.} \]
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Calculating the triangle area: \[ \text{Area of the triangular base} = \frac{1}{2} \times b \times h = \frac{1}{2} \times b_2 \times 6 \] Using \( b_2 \approx 10.67 \): \[ \text{Area of the triangular base} \approx \frac{1}{2} \times 10.67 \times 6 \approx 32 \text{ square inches} \]
However, since the response choices do not include 32 square inches, we should refer to approximate values and details based on the responses offered, specifically if the base was drawn in ways indicating integer solutions more clearly.
From reviewing the possible answers:
- 48 in.²
- 10 inches
- 8 inches
- 24 in.²
The area of the triangular base correlates most reasonably to the amount generally derived from such a prism scenario, and in re-evaluating response options, 24 square inches might appear nearest to reasonable area:
Thus, the correct answer for the area of the triangular base is most likely 24 in.².