Solving Problems with Volume Formulas - Geometry

A solid right cone and a solid cylinder have the same radius $r=3\text{ m}$ and the same height $h=8\text{ m}$ . Which claim about the volume is NOT justified?
Answer: The cone's volume equals the cylinder's volume because they share $r$ and $h$ . This problem asks which claim about cone and cylinder volumes is NOT justified. Both solids have radius r = 3 m and height h = 8 m. The cylinder volume formula is V = πr²h, and the cone volume formula is V = (1/3)πr²h. Since the cone formula includes the factor 1/3, the cone's volume is exactly one-third of the cylinder's volume when they share the same radius and height. Therefore, the cone and cylinder do NOT have equal volumes. Option D incorrectly claims they have equal volumes, making it the unjustified statement. Understanding the relationship between cone and cylinder volumes helps avoid calculation errors.
An ice cream cone is a right circular cone with radius $4\text{ cm}$ and height $9\text{ cm}$ . Which expression represents the volume of the cone?
Answer: $\frac{1}{3}\pi(4^2)(9)$ This problem asks for the volume expression of a cone-shaped ice cream cone. The solid is a right circular cone with radius 4 cm and height 9 cm. The volume formula for a cone is V = (1/3)πr²h, where r is the radius and h is the height. The correct expression is (1/3)π(4²)(9), which represents one-third of the volume of a cylinder with the same base and height. This formula accounts for the cone's tapering shape from base to apex. A common mistake is using the cylinder formula π(4²)(9), forgetting the factor of 1/3. Remember that a cone's volume is always one-third of a cylinder with the same dimensions.
A scoop of ice cream is shaped like a hemisphere with radius $4\text{ cm}$ . What is the volume of the solid?
Answer: $\frac{128}{3}\pi\text{ cm}^3$ This problem requires finding the volume of a hemisphere (half-sphere) of ice cream. The solid is a hemisphere with radius 4 cm. The volume formula for a sphere is V = (4/3)πr³, so a hemisphere has half this volume: V = (1/2)(4/3)πr³ = (2/3)πr³. Applying the formula: V = (2/3)π(4)³ = (2/3)π(64) = (128/3)π cm³. The volume is (128/3)π cubic centimeters. Option B doubles the correct answer, possibly confusing hemisphere with full sphere. When dealing with partial solids like hemispheres, adjust the standard formula by the appropriate fraction.
A right circular cone has radius $6\text{ cm}$ and height $9\text{ cm}$ . What is the volume of the cone?
Answer: $108\pi\text{ cm}^3$ Solving problems with volume formulas involves calculating the space occupied by three-dimensional solids using appropriate mathematical expressions. The solid in this problem is a right circular cone. The correct volume formula for a cone is $V = \frac{1}{3} \pi r^2 h$ , where r is the radius and h is the height. Applying the formula with r = 6 cm and h = 9 cm gives $V = \frac{1}{3} \pi (6)^2 (9) = 108 \pi \text{ cm}^3$ , matching choice B. This result correctly represents the cone's volume by including the one-third factor. A common distractor misconception is halving the full calculation, as in choice C, which might confuse with other formulas. To transfer this strategy, always identify the solid as a cone before selecting and applying the volume formula.
A conical paper cup has radius $3\text{ in}$ and height $10\text{ in}$ . What is the volume of the solid?
Answer: $30\pi\text{ in}^3$ This problem requires finding the volume of a conical paper cup. The solid is a cone with radius 3 inches and height 10 inches. The volume formula for a cone is V = (1/3)πr²h, where r is the radius and h is the height. Applying the formula: V = (1/3)π(3)²(10) = (1/3)π(9)(10) = (1/3)π(90) = 30π in³. The volume of the conical cup is 30π cubic inches. Option D shows 30 in³ without π, which is a common error when students forget that π is part of the answer. Always identify whether the solid has circular cross-sections to determine if π appears in the formula.
A cylindrical candle has radius $2.5\text{ in}$ and height $8\text{ in}$ . Which expression represents the volume of the candle?
Answer: $\pi(2.5)^2(8)$ Solving problems with volume formulas involves calculating the space occupied by three-dimensional solids using appropriate mathematical expressions. The solid in this problem is a right circular cylinder. The correct volume formula for a cylinder is V = πr²h, where r is the radius and h is the height. Applying the formula with r = 2.5 in and h = 8 in gives the expression π(2.5)²(8), matching choice A. This expression accurately computes the volume based on the cylindrical dimensions. A common distractor misconception is using the lateral surface area formula, as in choice B, which calculates 2πr h instead of volume. To transfer this strategy, always identify the solid as a cylinder before selecting and applying the volume formula.
A cylindrical water tank has an inside radius of $3\text{ m}$ and an inside height of $10\text{ m}$ . What is the volume of water the tank can hold when full?
Answer: $90\pi\text{ m}^3$ Solving problems with volume formulas involves calculating the space occupied by three-dimensional solids using appropriate mathematical expressions. The solid in this problem is a right circular cylinder. The correct volume formula for a cylinder is $V = \pi r^2 h$ , where r is the radius and h is the height. Applying the formula with $r = 3 \text{ m}$ and $h = 10 \text{ m}$ gives $V = \pi (3)^2 (10) = 90\pi \text{ m}^3$ . This result represents the maximum capacity of water the tank can hold when full, matching choice B. A common distractor misconception is using the surface area formula, such as choice D, which calculates $2\pi r h + 2\pi r^2$ instead of volume. To transfer this strategy, always identify the solid as a cylinder before selecting and applying the volume formula.
A cylindrical water tank has radius $3\text{ m}$ and height $10\text{ m}$ . What is the volume of the tank in cubic meters?
Answer: $90\pi\text{ m}^3$ This problem requires finding the volume of a cylindrical water tank. The solid is a right circular cylinder with radius 3 m and height 10 m. The volume formula for a cylinder is V = πr²h, where r is the radius and h is the height. Applying the formula: V = π(3)²(10) = π(9)(10) = 90π cubic meters. This result represents the total capacity of the cylindrical tank. A common error would be using diameter instead of radius, which would give π(6)²(10) = 360π, or forgetting to square the radius, yielding π(3)(10) = 30π. When solving volume problems, always identify the shape first to select the correct formula.
A cylindrical container and a spherical container are compared. The cylinder has radius $4\text{ cm}$ and height $12\text{ cm}$ . The sphere has radius $4\text{ cm}$ . Which value represents the total volume of both containers combined?
Answer: $\pi(4)^2(12)+\frac{4}{3}\pi(4)^3\text{ cm}^3$ The skill involves solving volume problems for geometric solids. The solids are a cylinder and a sphere combined. The volume is the sum of the cylinder's volume πr²h and the sphere's (4/3)πr³. Substituting r = 4 cm and h = 12 cm for the cylinder, and r = 4 cm for the sphere, gives π(4)²(12) + (4/3)π(4)³ cm³. This result matches choice A, providing the total volume. A common distractor is choice C, which adds surface areas instead of volumes. To transfer this strategy, always identify the solid before calculating its volume.
A candle is shaped like a cylinder with radius $2\text{ cm}$ and height $15\text{ cm}$ . A cylindrical hole of radius $0.5\text{ cm}$ is drilled straight through the center along the full height. Which value represents the total volume of wax remaining?
Answer: $\pi(2^2)(15) - \pi(0.5^2)(15)$ This problem involves finding the volume of a candle with a hole drilled through it. The solid is a cylinder with a cylindrical hole removed, where the outer cylinder has radius 2 cm and the inner hole has radius 0.5 cm, both with height 15 cm. To find the remaining volume, subtract the hole's volume from the original cylinder's volume: V = π(2²)(15) - π(0.5²)(15). This represents the volume of the outer cylinder minus the volume of the inner cylindrical hole. A common error is subtracting the radii first, calculating π((2-0.5)²)(15) = π(1.5²)(15), which incorrectly gives the volume of a solid cylinder with radius 1.5 cm. When dealing with hollow cylinders, always calculate volumes separately then subtract.
A solid metal sphere has radius $5\text{ cm}$ . What is the volume of the sphere?
Answer: $\frac{4}{3}\pi(5)^3\text{ cm}^3$ Solving problems with volume formulas involves calculating the space occupied by three-dimensional solids using appropriate mathematical expressions. The solid in this problem is a sphere. The correct volume formula for a sphere is $V = \frac{4}{3} \pi r^3$ , where $r$ is the radius. Applying the formula with $r = 5 \text{ cm}$ gives $V = \frac{4}{3} \pi (5)^3 \text{ cm}^3$ , matching choice A. This result accurately represents the volume by integrating the spherical geometry. A common distractor misconception is using the surface area formula, as in choice B, which calculates $4 \pi r^2$ instead of volume. To transfer this strategy, always identify the solid as a sphere before selecting and applying the volume formula.
A square pyramid has base side length $8\text{ ft}$ and height $15\text{ ft}$ . What is the volume of the solid?
Answer: $320\text{ ft}^3$ This problem requires finding the volume of a square pyramid. The solid is a pyramid with a square base of side length 8 ft and height 15 ft. The volume formula for a pyramid is V = (1/3)Bh, where B is the area of the base and h is the height. The base area is B = 8² = 64 ft², so V = (1/3)(64)(15) = (1/3)(960) = 320 ft³. The volume of the pyramid is 320 cubic feet. Option B shows 960 ft³, which would be the volume if students forgot to multiply by 1/3. When calculating pyramid volumes, remember to multiply by the fraction 1/3.
A concrete pillar is a cylinder with diameter $12\text{ ft}$ and height $5\text{ ft}$ . What is the volume of the solid?
Answer: $180\pi\text{ ft}^3$ This problem requires finding the volume of a concrete pillar. The solid is a cylinder with diameter 12 ft and height 5 ft. The volume formula for a cylinder is V = πr²h, but we must first convert diameter to radius: r = 12/2 = 6 ft. Applying the formula: V = π(6)²(5) = π(36)(5) = 180π ft³. The volume represents the amount of concrete needed to form the pillar. A common mistake is using diameter directly in the formula instead of radius, which would give π(12)²(5) = 720π ft³. Always convert diameter to radius by dividing by 2 before applying cylinder volume formulas.
A cone and a cylinder have the same base radius and height. If the volume of the cylinder is $432\pi$ cubic centimeters, what is the volume of the cone?
Answer: $144\pi$ cubic centimeters The volume of a cylinder is $V_{cylinder} = \pi r^2 h$ and the volume of a cone is $V_{cone} = \frac{1}{3}\pi r^2 h$ . Since they have the same base radius and height, $V_{cone} = \frac{1}{3} V_{cylinder} = \frac{1}{3} \times 432\pi = 144\pi$ cubic centimeters. Choice A ( $72\pi$ ) represents $\frac{1}{6}$ of the cylinder volume. Choice C ( $216\pi$ ) represents $\frac{1}{2}$ of the cylinder volume. Choice D ( $324\pi$ ) represents $\frac{3}{4}$ of the cylinder volume.
A square pyramid has a base side length of $8\text{ m}$ and height $15\text{ m}$ . What is the volume of the pyramid in cubic meters?
Answer: $320\text{ m}^3$ This problem requires finding the volume of a square pyramid. The solid is a square pyramid with base side length 8 m and height 15 m. The volume formula for a pyramid is V = (1/3)Bh, where B is the base area and h is the height. For a square base, B = s² = 8² = 64 m². Applying the formula: V = (1/3)(64)(15) = (1/3)(960) = 320 cubic meters. A common error is forgetting the factor of 1/3, which would give 960 m³ instead. When solving pyramid problems, remember that the volume is always one-third of the base area times the height.
A cylindrical grain silo has radius $7\text{ ft}$ and height $20\text{ ft}$ . Which claim about the volume is NOT justified?
Answer: The volume is $2\pi(7)(20)$ cubic feet. The skill involves solving problems with volume formulas. The solid is a cylinder. The volume of a cylinder is given by the formula V = π r² h. Substituting r = 7 ft and h = 20 ft confirms that V = π (7²) (20) or equivalently 980π cubic feet. Claims A and B correctly apply this, and D states a true property since volume scales with r². Choice C confuses volume with lateral surface area, misapplying 2π r h. To apply this to other problems, always identify the solid before selecting the volume formula.
A right circular cone has radius $6\text{ ft}$ and height $8\text{ ft}$ . What is the volume of the cone in cubic feet?
Answer: $96\pi\text{ ft}^3$ This problem requires finding the volume of a right circular cone. The solid is a cone with radius 6 ft and height 8 ft. The volume formula for a cone is V = (1/3)πr²h, where r is the radius and h is the height. Applying the formula: V = (1/3)π(6)²(8) = (1/3)π(36)(8) = (1/3)π(288) = 96π cubic feet. This represents one-third of the volume of a cylinder with the same dimensions. A common error is omitting the factor of 1/3, which would give 288π ft³. When calculating cone volume, always remember to multiply by 1/3.
A cylindrical juice can has radius $3\text{ cm}$ and height $12\text{ cm}$ . Which claim about the volume is NOT justified?
Answer: The volume is $2\pi(3)(12)\text{ cm}^3$ . Solving problems with volume formulas involves calculating the space occupied by three-dimensional solids using appropriate mathematical expressions. The solid in this problem is a right circular cylinder. The correct volume formula for a cylinder is V = $\pi r^2 h$ , where r is the radius and h is the height. Applying the formula with r = 3 cm and h = 12 cm gives V = $\pi (3)^2 (12) = 108\pi$ cm³, which shows claims A and B are justified while D is a valid property. However, claim C uses $2\pi r h$ , the lateral surface area, incorrectly as volume, making it not justified and matching choice C. A common distractor misconception is confusing volume with surface area formulas, as seen in choice C. To transfer this strategy, always identify the solid as a cylinder before evaluating claims about its volume.
A right circular cone has radius $5\text{ cm}$ and height $12\text{ cm}$ . Which expression represents the volume of the cone?
Answer: $\frac{1}{3}\pi(5)^2(12)\text{ cm}^3$ The skill involves solving volume problems for geometric solids. The solid is a right circular cone. The volume formula for a cone is V = $\frac{1}{3} \pi r^2 h$ . Substituting the radius of 5 cm and height of 12 cm gives $\frac{1}{3} \pi (5)^2 (12) \text{ cm}^3$ . This result matches choice C, correctly representing the volume. A common distractor is choice A, which uses the cylinder formula without the 1/3 factor. To transfer this strategy, always identify the solid before calculating its volume.
A right circular cone and a cylinder have the same base radius of $5\text{ cm}$ and the same height of $12\text{ cm}$ . Which claim about the volume is NOT justified?
Answer: The cone's volume is $\pi(5^2)(12)$ cubic centimeters. This problem asks which claim about cone and cylinder volumes is NOT justified. The solids are a cone and cylinder with the same base radius (5 cm) and height (12 cm). For the cylinder, $V = \pi r^2 h = \pi(5^2)(12)$ cubic centimeters, and for the cone, $V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi(5^2)(12)$ cubic centimeters. The cone's volume is indeed one-third of the cylinder's volume, both volumes are measured in $\text{cm}^3$ , and the cylinder's volume is $\pi(5^2)(12)$ $\text{cm}^3$ . However, the cone's volume is $\frac{1}{3} \pi(5^2)(12)$ , not $\pi(5^2)(12)$ cubic centimeters. The unjustified claim confuses the cone formula with the cylinder formula by omitting the factor of 1/3. Always remember that cone volume requires the one-third factor.

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