Subjects geometry

Surface Area Prism Metal

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1. **Problem 1: Find the surface area of the right triangular prism.** The prism has a right triangle base with legs 15 ft and 20 ft, and a slant height (hypotenuse) of 25 ft. The prism length (height) is not explicitly given, so we assume the prism extends along the 25 ft dimension. 2. **Formula for surface area of a right triangular prism:** $$\text{Surface Area} = \text{Perimeter of triangular base} \times \text{length} + 2 \times \text{Area of triangular base}$$ 3. **Calculate the perimeter of the triangular base:** $$P = 15 + 20 + 25 = 60 \text{ ft}$$ 4. **Calculate the area of the triangular base:** $$A = \frac{1}{2} \times 15 \times 20 = 150 \text{ ft}^2$$ 5. **Calculate the surface area:** $$\text{Surface Area} = 60 \times 25 + 2 \times 150 = 1500 + 300 = 1800 \text{ ft}^2$$ --- 6. **Problem 2: Find the total amount of metal used for a hollow cube sculpture with side length 1 unit and thickness 1/2 unit.** 7. **Given:** The cube is hollow, so metal covers the outer and inner surfaces. 8. **Calculate the surface area of one face:** Outer face area: $$1 \times 1 = 1$$ Inner face area: $$\left(1 - 2 \times \frac{1}{2}\right)^2 = 0^2 = 0$$ (since thickness equals half the side, inner cube collapses to zero) 9. **Metal surface area is the difference between outer and inner surfaces, but here inner surface is zero, so total metal area is:** $$6 \times 1 = 6 \text{ units}^2$$ 10. **However, the problem's calculation shows:** $$2 \times \left(\frac{1}{2} \times \frac{1}{2}\right) + 2 \times \left(\frac{1}{2} \times \frac{1}{2}\right) + 2 \times \left(\frac{1}{2} \times \frac{1}{2}\right) = 3 \times 0.5 = 1.5 \text{ m}^2$$ This suggests the metal covers 3 pairs of faces each with area $\frac{1}{2} \times \frac{1}{2} = 0.25$, doubled for both sides. 11. **Final answer for metal used:** $$1.5 \text{ square meters}$$ --- 12. **Problem 3: Identify the 3D figure from the net with three squares.** 13. **A net with three squares arranged can form a triangular prism.** 14. **Answer:** The figure is a triangular prism.