This question tests applying volume formulas: cylinder $V=\pi r^2 h$, cone $V=\frac{1}{3} \pi r^2 h$, sphere $V=\frac{4}{3} \pi r^3$, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area $\pi r^2$ times height h giving $V=\pi r^2 h$ (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving $V=\frac{1}{3} \pi r^2 h$ (tapers to point reducing volume to one-third), sphere is $V=\frac{4}{3} \pi r^3$ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: $r=d/2$), calculate (exponents first, multiply, approximate $\pi \approx 3.14$ or leave exact). For cone with d=8 cm so r=4 cm, h=6 cm, $V=\frac{1}{3} \pi(4^2)(6)=\frac{1}{3} \pi(16)(6)=32 \pi \text{ cm}^3$, matching C. Common errors: using diameter as r for 128$\pi$, cylinder formula for 96$\pi$, sphere for $\frac{256}{3} \pi$, or missing 1/3. Steps: (1) identify cone, (2) gather r=4 cm from d=8, h=6 cm, (3) select $\frac{1}{3} \pi r^2 h$, (4) substitute, (5) calculate 32$\pi$, (6) add cm$^3$. Halving diameter is crucial for accurate radius.

This question tests applying volume formulas: cylinder V=πr²h, cone V=(1/3)πr²h, sphere V=(4/3)πr³, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area πr² times height h giving V=πr²h (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving V=(1/3)πr²h (tapers to point reducing volume to one-third), sphere is V=(4/3)πr³ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: r=d/2), calculate (exponents first, multiply, approximate π≈3.14 or leave exact). For this sphere with d=10 ft so r=5 ft, V=(4/3)π(5³)=(4/3)π(125)=500/3 π ft³, matching choice B. Common errors include using diameter as radius for 4000/3 π, applying cylinder formula like π(5²)(10), or omitting height incorrectly. Steps: (1) identify sphere, (2) gather r=5 ft from d=10, (3) select V=(4/3)πr³, (4) substitute value, (5) calculate 500/3 π, (6) add ft³. Always halve diameter for radius in sphere calculations.

This question tests applying volume formulas: cylinder V=πr²h, cone V=(1/3)πr²h, sphere V=(4/3)πr³, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area πr² times height h giving V=πr²h (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving V=(1/3)πr²h (tapers to point reducing volume to one-third), sphere is V=(4/3)πr³ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: r=d/2), calculate (exponents first, multiply, approximate π≈3.14 or leave exact). For this cylinder with r=3 cm and h=10 cm, V=π(3²)(10)=π(9)(10)=90π cm³, matching choice B. Common errors include using diameter instead of radius, wrong exponent like r³ for cylinder, arithmetic mistakes such as 3²=6, or forgetting cubic units. Steps: (1) identify cylinder, (2) gather r=3 cm, h=10 cm, (3) select V=πr²h, (4) substitute values, (5) calculate 90π, (6) add cm³. Remember, leaving in terms of π keeps it exact, as required here.

This question tests applying volume formulas: cylinder V=πr²h, cone V=(1/3)πr²h, sphere V=(4/3)πr³, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area πr² times height h giving V=πr²h (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving V=(1/3)πr²h (tapers to point reducing volume to one-third), sphere is V=(4/3)πr³ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: r=d/2), calculate (exponents first, multiply, approximate π≈3.14 or leave exact). For this cylinder with r=2 m and h=5 m, V=π(2²)(5)=π(4)(5)=20π m³, matching choice B. Common errors include using cone formula for 20/3 π, doubling radius incorrectly, arithmetic like 2²=2, or non-cubic units. Steps: (1) identify cylinder, (2) gather r=2 m, h=5 m, (3) select V=πr²h, (4) substitute values, (5) calculate 20π, (6) add m³. This represents the tank's capacity in exact terms with π.

This question tests applying volume formulas: cylinder $V=\pi r^2 h$, cone $V=\frac{1}{3} \pi r^2 h$, sphere $V=\frac{4}{3} \pi r^3$, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area $\pi r^2$ times height h giving $V=\pi r^2 h$ (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving $V=\frac{1}{3} \pi r^2 h$ (tapers to point reducing volume to one-third), sphere is $V=\frac{4}{3} \pi r^3$ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: $r=d/2$), calculate (exponents first, multiply, approximate $\pi\approx3.14$ or leave exact). For $V=288\pi$, $(4/3)\pi r^3=288\pi$ so $r^3=288\times(3/4)=216$, $r=6 \text{ m}$ ($6^3=216$), matching B. Common errors: ignoring 4/3, getting r=9 (wrong inverse), or confusing with other formulas. Steps: (1) identify sphere, (2) set $(4/3)\pi r^3=288\pi$, (3) solve $r^3=216$, (4) cube root to r=6, (5) check units m. Reverse calculations test formula understanding.

This question tests applying volume formulas: cylinder V=πr²h, cone V=(1/3)πr²h, sphere V=(4/3)πr³, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area πr² times height h giving V=πr²h (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving V=(1/3)πr²h (tapers to point reducing volume to one-third), sphere is V=(4/3)πr³ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: r=d/2), calculate (exponents first, multiply, approximate π≈3.14 or leave exact). Cylinder V=π(3²)(4)=36π, sphere V=(4/3)π(3³)=(4/3)π(27)=36π, so equal, matching C. Common errors: wrong formulas like cone for cylinder, or miscalculating r³=27. Steps: (1) identify shapes, (2) gather dimensions, (3) select formulas, (4) calculate both, (5) compare 36π=36π, (6) conclude equal. Comparisons require computing each volume fully.

This question tests applying volume formulas: cylinder V=πr²h, cone V=(1/3)πr²h, sphere V=(4/3)πr³, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area πr² times height h giving V=πr²h (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving V=(1/3)πr²h (tapers to point reducing volume to one-third), sphere is V=(4/3)πr³ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: r=d/2), calculate (exponents first, multiply, approximate π≈3.14 or leave exact). For this sphere with r=6 m, V=(4/3)π(6³)=(4/3)π(216)=288π m³, matching choice B. Common errors include using r² instead of r³ (like 36π times something), forgetting the 4/3 factor for 216π, dividing instead of multiplying, or incorrect cubing like 6³=216. Steps: (1) identify sphere, (2) gather r=6 m, (3) select V=(4/3)πr³, (4) substitute value, (5) calculate 288π, (6) add m³. Spheres use only radius, no height, due to their uniform symmetry.

This question tests applying volume formulas: cylinder V=πr²h, cone V=(1/3)πr²h, sphere V=(4/3)πr³, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area πr² times height h giving V=πr²h (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving V=(1/3)πr²h (tapers to point reducing volume to one-third), sphere is V=(4/3)πr³ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: r=d/2), calculate (exponents first, multiply, approximate π≈3.14 or leave exact). The correct cone formula is V=(1/3)πr²h, matching choice C, as it accounts for the tapering volume. Common errors include confusing with cylinder (no 1/3), sphere (r³), or made-up like πr³h. Steps: (1) identify cone by its point and base, (2) recall need for 1/3 factor, (3) select (1/3)πr²h, (4) distinguish from others. Memorize formulas to avoid mixing cone and cylinder.

This question tests applying volume formulas: cylinder $V=\pi r^2 h$, cone $V=\frac{1}{3} \pi r^2 h$, sphere $V=\frac{4}{3} \pi r^3$, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area $\pi r^2$ times height $h$ giving $V=\pi r^2 h$ (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving $V=\frac{1}{3} \pi r^2 h$ (tapers to point reducing volume to one-third), sphere is $V=\frac{4}{3} \pi r^3$ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: $r=d/2$), calculate (exponents first, multiply, approximate $\pi \approx 3.14$ or leave exact). The correct cylinder formula is $V=\pi r^2 h$, matching choice B, representing base area times height. Common errors include adding 1/3 like cone, using $r^3$ like sphere, or surface area like $2 \pi r h$. Steps: (1) identify cylinder by uniform height, (2) recall $\pi r^2 h$, (3) select from options, (4) avoid extras like fractions unless tapered. Formulas derive from geometry, so visualize shapes.

This question tests applying volume formulas: cylinder V=πr²h, cone V=(1/3)πr²h, sphere V=(4/3)πr³, selecting correct formula for shape and calculating accurately. Each shape has specific formula based on geometry: cylinder is base area πr² times height h giving V=πr²h (circular cross-section throughout height), cone is 1/3 of cylinder with same base and height giving V=(1/3)πr²h (tapers to point reducing volume to one-third), sphere is V=(4/3)πr³ (radius cubed, no height—symmetric). Apply: identify shape, use corresponding formula with given dimensions (convert diameter to radius if needed: r=d/2), calculate (exponents first, multiply, approximate π≈3.14 or leave exact). For a cylinder, the correct formula is V=πr²h, based on base area times height. Common errors include selecting cone formula (with 1/3), sphere (with r³), or invented ones like πr³h. Steps: (1) identify shape as cylinder, (2) recall need for r and h, (3) choose πr²h from options, (4) verify against geometry (no 1/3, no r³), (5) distinguish from similar shapes, (6) memorize for application. Understanding derivations helps select the right formula without confusion.