The triangular base has area ½ × 5 × 12 = 30 sq cm. The smaller piece has the same base area but height of 3 cm. Volume = base area × height = 30 × 3 = 90 cubic cm. Choice B uses height of 4 cm (8-4 error). Choice C uses height of 5 cm (leg length confusion). Choice D uses height of 6 cm (8-2 error).

This problem tests solving area problems for composite figures by decomposing into simpler shapes (rectangles, triangles, prisms, pyramids), applying formulas, and combining results. Formulas: triangle A=(1/2)bh, rectangle A=lw, rectangular prism V=lwh, pyramid V=(1/3)Bh (B=base area), triangular prism V=((1/2)bh)×length (triangle base area times prism length). The cutout has: rectangle area = 8 × 4 = 32 cm², each triangle area = (1/2) × 4 × 3 = 6 cm², two triangles total = 2 × 6 = 12 cm², so total area = 32 + 12 = 44 cm². The correct total area is 44 cm². Common errors include forgetting the (1/2) in triangle area formula (using 4 × 3 = 12 per triangle, giving total 56), or counting only one triangle instead of two. Steps: (1) identify composite structure (rectangle plus two triangles), (2) calculate rectangle area (8 × 4 = 32), (3) calculate one triangle area using A = (1/2)bh = (1/2) × 4 × 3 = 6, (4) multiply by 2 for two congruent triangles (2 × 6 = 12), (5) add all areas (32 + 12 = 44), (6) verify units (cm²).

This question tests solving area, volume, and surface area problems for composite figures by decomposing into simpler shapes (rectangles, triangles, prisms, pyramids), applying formulas, and combining results. Composite figures: decompose into standard shapes (L-shape as two rectangles: 10×5=50 and 6×3=18, sum: 68; or as large minus cutout: 10×8=80 minus 4×3=12, difference: 68, equivalent). Formulas: triangle A=(1/2)bh, rectangle A=lw, rectangular prism V=lwh, pyramid V=(1/3)Bh (B=base area), triangular prism V=((1/2)bh)×length (triangle base area times prism length). Surface area: sum all face areas (rectangular prism 3×4×5 has faces: two 3×4=12, two 3×5=15, two 4×5=20, total: 2(12+15+20)=94). For this model, decompose into prism 5 cm × 4 cm × 6 cm = 120 cm³ and pyramid with base 5 cm × 4 cm = 20 cm², height 3 cm, V=(1/3)×20×3=20 cm³, total 140 cm³. Common errors include forgetting the (1/3) for pyramid (using 60 cm³, total 180 cm³), arithmetic mistake (120+20=130 cm³), or treating pyramid as prism (60 cm³ total 180 cm³). Steps: (1) identify composite structure (prism with pyramid top), (2) decompose into standard shapes (prism and pyramid), (3) calculate each component (apply formulas: V=lwh, V=(1/3)Bh), (4) combine (add volumes), (5) verify units (volume cm³).

In 3 hours, the water level drops 2 × 3 = 6 feet. The volume of water removed is a cylinder with radius 6 feet and height 6 feet: V = πr²h = π(6²)(6) = 216π cubic feet. Choice B uses height of 10 instead of 6. Choice C doubles the correct answer. Choice D uses the total tank volume incorrectly.

This problem tests solving area problems for composite figures by decomposing into simpler shapes (rectangles, triangles, prisms, pyramids), applying formulas, and combining results. Formulas: triangle A=(1/2)bh, rectangle A=lw, rectangular prism V=lwh, pyramid V=(1/3)Bh (B=base area), triangular prism V=((1/2)bh)×length (triangle base area times prism length). The house-shaped display has: rectangle area = 8 × 5 = 40 in², triangle area = (1/2) × 8 × 3 = 12 in², so total area = 40 + 12 = 52 in². The correct total area is 52 in². Common errors include forgetting the (1/2) in the triangle formula (using 8 × 3 = 24, giving total 64), or arithmetic mistakes in addition. Steps: (1) identify composite structure (rectangle with triangle on top), (2) calculate rectangle area (8 × 5 = 40), (3) calculate triangle area using A = (1/2)bh = (1/2) × 8 × 3 = 12, (4) add areas (40 + 12 = 52), (5) verify units (in²). The "house" shape is a common composite figure—remember the triangle on top uses the same base width as the rectangle below.

This problem tests solving volume problems for composite figures by decomposing into simpler shapes (rectangles, triangles, prisms, pyramids), applying formulas, and combining results. Formulas: triangle A=(1/2)bh, rectangle A=lw, rectangular prism V=lwh, pyramid V=(1/3)Bh (B=base area), triangular prism V=((1/2)bh)×length (triangle base area times prism length). For the triangular prism: triangular base area = (1/2) × 9 × 4 = 18 m², volume = 18 × 5 = 90 m³. The correct volume is 90 m³. Common errors include forgetting the (1/2) in the triangle area formula (using 9 × 4 = 36, giving volume 180), or confusing the prism length with other dimensions. Steps: (1) identify the shape (triangular prism), (2) calculate triangular base area using A = (1/2)bh = (1/2) × 9 × 4 = 18 m², (3) multiply base area by prism length: V = 18 × 5 = 90 m³, (4) verify units (m³). The triangular prism volume formula is (triangular base area) × length—don't forget the (1/2) factor in the triangle area.

This question tests solving area, volume, and surface area problems for composite figures by decomposing into simpler shapes (rectangles, triangles, prisms, pyramids), applying formulas, and combining results. Composite figures: decompose into standard shapes (L-shape as two rectangles: 10×5=50 and 6×3=18, sum: 68; or as large minus cutout: 10×8=80 minus 4×3=12, difference: 68, equivalent). Formulas: triangle A=(1/2)bh, rectangle A=lw, rectangular prism V=lwh, pyramid V=(1/3)Bh (B=base area), triangular prism V=((1/2)bh)×length (triangle base area times prism length). Surface area: sum all face areas (rectangular prism 3×4×5 has faces: two 3×4=12, two 3×5=15, two 4×5=20, total: 2(12+15+20)=94). For this L-shaped floor, decompose as large rectangle 10 m × 8 m = 80 m² minus cutout 4 m × 3 m = 12 m², resulting in 68 m²; alternatively, two rectangles: one 10 m × 5 m = 50 m² and one 6 m × 3 m = 18 m² (assuming the cutout leaves an L with those dimensions), total 68 m². Common errors include calculating the large rectangle only (80 m²), adding instead of subtracting the cutout (92 m²), or wrong decomposition like treating as single shape without adjustment. Steps: (1) identify composite structure (L-shape with cutout), (2) decompose into standard shapes (large rectangle minus small rectangle), (3) calculate each component (apply formulas: A=lw), (4) combine (subtract cutout), (5) verify units (area m²). Decomposition choice: two rectangles OR large-minus-small (both valid, should give same answer—good check).

Tests solving area, volume, and surface area problems for composite figures by decomposing into simpler shapes (rectangles, triangles, prisms, pyramids), applying formulas, and combining results. Composite figures: decompose into standard shapes (L-shape as two rectangles: 10×5=50 and 6×3=18, sum: 68; or as large minus cutout: 10×8=80 minus 4×3=12, difference: 68, equivalent). Formulas: triangle A=(1/2)bh, rectangle A=lw, rectangular prism V=lwh, pyramid V=(1/3)Bh (B=base area), triangular prism V=((1/2)bh)×length (triangle base area times prism length). For this triangular prism, first find the triangular base area: A=(1/2)×6×4=12 cm², then multiply by prism length: V=12×10=120 cm³. Common error would be forgetting the (1/2) in the triangle area formula, using 6×4=24 instead of 12, giving volume 240 cm³. Steps: (1) identify shape (triangular prism), (2) calculate triangular base area using A=(1/2)bh=(1/2)×6×4=12 cm², (3) multiply by prism length for volume V=12×10=120 cm³, (4) verify units (volume in cm³). The key is remembering that triangular prism volume equals (triangular base area)×(prism length), and the triangular area requires the factor (1/2).

This problem tests solving volume problems for composite figures by decomposing into simpler shapes (rectangles, triangles, prisms, pyramids), applying formulas, and combining results. Formulas: triangle A=(1/2)bh, rectangle A=lw, rectangular prism V=lwh, pyramid V=(1/3)Bh (B=base area), triangular prism V=((1/2)bh)×length (triangle base area times prism length). The toy block consists of two rectangular prisms: Prism 1 volume = 6 × 4 × 2 = 48 cm³, Prism 2 volume = 3 × 4 × 2 = 24 cm³, so total volume = 48 + 24 = 72 cm³. The correct total volume is 72 cm³. Common errors include arithmetic mistakes in calculating individual volumes or in the final addition, or misunderstanding that the volumes should be added (not multiplied). Steps: (1) identify composite structure (two rectangular prisms), (2) calculate Prism 1 volume (6 × 4 × 2 = 48), (3) calculate Prism 2 volume (3 × 4 × 2 = 24), (4) add volumes since they don't overlap (48 + 24 = 72), (5) verify units (cm³). When prisms are stacked without overlap, their volumes simply add together.

Tests solving area, volume, and surface area problems for composite figures by decomposing into simpler shapes (rectangles, triangles, prisms, pyramids), applying formulas, and combining results. Composite figures: decompose into standard shapes (L-shape as two rectangles: 10×5=50 and 6×3=18, sum: 68; or as large minus cutout: 10×8=80 minus 4×3=12, difference: 68, equivalent). Formulas: triangle A=(1/2)bh, rectangle A=lw, rectangular prism V=lwh, pyramid V=(1/3)Bh (B=base area), triangular prism V=((1/2)bh)×length (triangle base area times prism length). For this right triangular prism, the triangular base area is A=(1/2)×3×8=12 cm² (using legs as base and height), then volume is V=12×7=84 cm³. Common error would be forgetting the (1/2) factor, using 3×8=24 for base area, giving volume 168 cm³. Steps: (1) identify shape (right triangular prism), (2) calculate triangular base area using A=(1/2)×leg₁×leg₂=(1/2)×3×8=12 cm², (3) multiply by prism length for volume V=12×7=84 cm³, (4) verify units (volume in cm³). For right triangular prisms, the legs of the right triangle serve as base and height in the area formula.