ISEE Upper Level Mathematics Achievement Quiz: 3 D Volume
20 questions · exam conditions
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3 D VolumeQuestion 1 of 20

A hexagonal prism has a base with an area of 36 square meters and a height of 8 meters. What is the volume of the prism?

144 cubic meters
216 cubic meters
288 cubic meters
432 cubic meters
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ISEE Upper Level Mathematics Achievement Quiz

ISEE Upper Level Mathematics Achievement Quiz: 3 D Volume

Practice 3 D Volume in ISEE Upper Level Mathematics Achievement with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on 3 D Volume, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level Mathematics Achievement.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A hexagonal prism has a base with an area of 36 square meters and a height of 8 meters. What is the volume of the prism?

  1. 144 cubic meters
  2. 216 cubic meters
  3. 288 cubic meters (correct answer)
  4. 432 cubic meters
Explanation: When you encounter prism volume problems, remember that the volume of any prism equals the area of its base multiplied by its height. The shape of the base doesn't matter - whether it's triangular, rectangular, hexagonal, or any other polygon, this formula always applies. Here, you have a hexagonal prism with a base area of 36 square meters and a height of 8 meters. Using the volume formula: V=Base Area×Height=36×8=288 cubic metersV = \text{Base Area} \times \text{Height} = 36 \times 8 = 288 \text{ cubic meters} Looking at the wrong answers reveals common calculation errors. Choice A (144 cubic meters) comes from accidentally dividing instead of multiplying: 36×8÷2=14436 \times 8 ÷ 2 = 144. This might happen if you confuse prism volume with triangle area formulas. Choice B (216 cubic meters) results from using 6 as the height instead of 8, possibly because you focused on the "hexa" prefix meaning six: 36×6=21636 \times 6 = 216. Choice D (432 cubic meters) comes from incorrectly multiplying by an extra factor, perhaps thinking you need to account for the hexagon having 6 sides: 36×8×1.5=43236 \times 8 \times 1.5 = 432. The key insight is that the base area is already given to you - you don't need to calculate anything about the hexagon's sides or angles. Whether the base is a triangle, square, or 20-sided polygon, if you're told the area, simply multiply by the height. This makes prism problems much more straightforward than they initially appear.

Question 2

A pyramid has a rectangular base measuring 12 feet by 9 feet and a height of 20 feet. What is its volume?

  1. 720 cubic feet (correct answer)
  2. 1,080 cubic feet
  3. 1,800 cubic feet
  4. 2,160 cubic feet
Explanation: When you encounter pyramid volume problems, remember that pyramids always have one-third the volume of a prism with the same base and height. This is a fundamental relationship that applies to all pyramids, regardless of their base shape. To find this pyramid's volume, you need the formula: V=13×base area×heightV = \frac{1}{3} \times \text{base area} \times \text{height}. First, calculate the rectangular base area: 12×9=10812 \times 9 = 108 square feet. Then multiply by the height and apply the one-third factor: V=13×108×20=21603=720V = \frac{1}{3} \times 108 \times 20 = \frac{2160}{3} = 720 cubic feet. Looking at the wrong answers: Choice B (1,080) likely comes from forgetting the one-third factor and instead using one-half, as if this were a triangular prism formula. Choice C (1,800) might result from incorrectly calculating the base area or making an arithmetic error in the final division. Choice D (2,160) is the result you'd get if you completely forgot the one-third factor and just multiplied base area times height—this would give you the volume of a rectangular prism, not a pyramid. The correct answer is A) 720 cubic feet. For pyramid problems, always double-check that you've included the one-third factor in your calculation. A helpful way to remember this: imagine filling a pyramid with sand, then pouring that sand into a box with the same base and height—you'd need exactly three pyramids' worth of sand to fill the box completely.

Question 3

A rectangular prism has dimensions 6 cm by 4 cm by 9 cm. If each dimension is doubled, by what factor does the volume increase?

  1. 2
  2. 4
  3. 6
  4. 8 (correct answer)
Explanation: When you encounter problems about scaling three-dimensional objects, remember that volume changes more dramatically than you might expect because it involves all three dimensions. Let's calculate the original volume first. For a rectangular prism, volume equals length × width × height: 6×4×9=216 cm36 \times 4 \times 9 = 216 \text{ cm}^3. When each dimension is doubled, the new dimensions become 12 cm by 8 cm by 18 cm. The new volume is 12×8×18=1,728 cm312 \times 8 \times 18 = 1,728 \text{ cm}^3. To find the factor of increase, divide the new volume by the original: 1,728216=8\frac{1,728}{216} = 8. Here's why each wrong answer represents a common misconception: Choice A (2) assumes the volume doubles because each dimension doubles—but this ignores that volume is three-dimensional. Choice B (4) might come from thinking that doubling two dimensions gives you four times the volume, forgetting about the third dimension. Choice C (6) could result from incorrectly adding the scaling factor (2) for each of the three dimensions: 2 + 2 + 2 = 6. Choice D (8) is correct because when you scale all three dimensions by a factor of 2, the volume scales by 23=82^3 = 8. Key strategy: For any scaling problem involving three-dimensional objects, remember that the volume scaling factor equals the linear scaling factor raised to the third power. If each dimension increases by factor kk, volume increases by factor k3k^3. This pattern appears frequently on geometry problems involving similar solids.

Question 4

A pyramid has a square base with side length 8 meters and a height of 15 meters. What is its volume?

  1. 160 cubic meters
  2. 320 cubic meters (correct answer)
  3. 480 cubic meters
  4. 960 cubic meters
Explanation: When you encounter a pyramid volume problem, remember that pyramids have one-third the volume of a prism with the same base and height. The formula is V=13×base area×heightV = \frac{1}{3} \times \text{base area} \times \text{height}. Since this pyramid has a square base with side length 8 meters, you first calculate the base area: 82=648^2 = 64 square meters. Then apply the pyramid volume formula: V=13×64×15=9603=320V = \frac{1}{3} \times 64 \times 15 = \frac{960}{3} = 320 cubic meters. Let's examine why the other answers are incorrect. Choice A (160 cubic meters) represents a calculation error where someone might have used 16\frac{1}{6} instead of 13\frac{1}{3}, perhaps confusing pyramid and cone formulas. Choice C (480 cubic meters) suggests someone calculated 23×64×15\frac{2}{3} \times 64 \times 15, incorrectly using two-thirds instead of one-third. Choice D (960 cubic meters) is what you'd get if you forgot the 13\frac{1}{3} factor entirely and calculated 64×1564 \times 15—this would be the volume of a rectangular prism, not a pyramid. The correct answer is B (320 cubic meters). For pyramid problems, always remember the key fraction: 13\frac{1}{3}. A helpful way to remember this is that pyramids "taper to a point," so they hold much less volume than a full rectangular box with the same base and height. Double-check that you're using one-third, not mistakenly calculating the full prism volume.

Question 5

A sphere has the same volume as a cube with edge length 12 inches. What is the radius of the sphere to the nearest tenth of an inch?

  1. 8.7 inches
  2. 9.3 inches (correct answer)
  3. 9.8 inches
  4. 10.4 inches
Explanation: When you encounter problems involving equal volumes between different shapes, you need to set up an equation using the volume formulas for each shape and solve for the unknown dimension. First, find the volume of the cube. With edge length 12 inches, the cube's volume is 123=1,72812^3 = 1,728 cubic inches. Since the sphere has the same volume, you can set up the equation using the sphere volume formula V=43πr3V = \frac{4}{3}\pi r^3: 43πr3=1,728\frac{4}{3}\pi r^3 = 1,728 Solving for the radius: r3=1,728×34π=5,1844π=1,296πr^3 = \frac{1,728 \times 3}{4\pi} = \frac{5,184}{4\pi} = \frac{1,296}{\pi} r3=1,2963.14159...≈412.7r^3 = \frac{1,296}{3.14159...} \approx 412.7 r=412.73≈9.3r = \sqrt[3]{412.7} \approx 9.3 inches Looking at the wrong answers: Choice A (8.7 inches) results from calculation errors, likely in the cube root step. Choice C (9.8 inches) might come from using an incorrect value of π or making arithmetic mistakes in the division. Choice D (10.4 inches) could result from forgetting the 43\frac{4}{3} coefficient in the sphere volume formula, using just πr3\pi r^3 instead. The correct answer is B) 9.3 inches. Study tip: For volume comparison problems, always write down both volume formulas clearly before substituting values. Double-check that you're using the complete formulas—sphere volume problems often trip students up with that 43\frac{4}{3} coefficient.

Question 6

A regular octagonal prism has a base area of 48 square centimeters and a volume of 384 cubic centimeters. What is the height of the prism?

  1. 6 centimeters
  2. 8 centimeters (correct answer)
  3. 12 centimeters
  4. 16 centimeters
Explanation: When you encounter a prism volume problem, remember that the volume formula is straightforward: V=base area×heightV = \text{base area} \times \text{height}. This relationship holds for any prism, regardless of the shape of its base. You're given that the octagonal prism has a base area of 48 square centimeters and a volume of 384 cubic centimeters. To find the height, substitute these values into the volume formula: 384=48×h384 = 48 \times h Solving for height: h=38448=8h = \frac{384}{48} = 8 centimeters Let's examine why the other answers are incorrect: (A) 6 centimeters would give a volume of 48×6=28848 \times 6 = 288 cubic centimeters, which is too small. (C) 12 centimeters would produce a volume of 48×12=57648 \times 12 = 576 cubic centimeters, which exceeds the given volume. (D) 16 centimeters would result in 48×16=76848 \times 16 = 768 cubic centimeters, which is exactly double the actual volume. The correct answer is (B) 8 centimeters. Study tip: Don't be intimidated by complex geometric shapes like octagons. When dealing with prisms, the base shape doesn't change the volume calculation—you always multiply base area by height. Focus on identifying what information you're given and what you need to find, then apply the basic formula. Practice recognizing when you can work backwards from the volume formula to find missing dimensions.

Question 7

Two cylinders have the same height. The first cylinder has a radius of 3 units, and the second has a radius of 6 units. What is the ratio of their volumes?

  1. 1:2
  2. 1:4 (correct answer)
  3. 2:1
  4. 4:1
Explanation: When you encounter problems comparing volumes of similar shapes with different dimensions, focus on how the volume formula responds to changes in each dimension. The volume of a cylinder is V=πr2hV = \pi r^2 h. Since both cylinders have the same height, you can factor that out and focus on how the radius affects volume. For the first cylinder: V1=π(3)2h=9πhV_1 = \pi (3)^2 h = 9\pi h. For the second cylinder: V2=π(6)2h=36πhV_2 = \pi (6)^2 h = 36\pi h. The ratio of their volumes is V1V2=9πh36πh=936=14\frac{V_1}{V_2} = \frac{9\pi h}{36\pi h} = \frac{9}{36} = \frac{1}{4}, which gives us 1:4. Choice A (1:2) incorrectly assumes volume changes linearly with radius. This would be true if volume were V=πrhV = \pi r h, but the radius is actually squared in the formula. Choice C (2:1) and choice D (4:1) both flip the ratio direction—these would suggest the smaller cylinder has greater volume than the larger one, which is impossible. Choice D specifically represents the misconception that the larger cylinder's volume is four times smaller rather than four times larger. Remember that when comparing volumes of similar 3D shapes, linear dimensions get raised to the power that matches the dimension count. Since volume is 3-dimensional, if one linear measurement doubles, volume increases by 23=82^3 = 8 times. Here, radius doubled from 3 to 6, so volume increased by 22=42^2 = 4 times (since only radius changed, not all dimensions).

Question 8

A triangular prism has a triangular base with an area of 24 square inches and a height of 10 inches. What is the volume of the prism?

  1. 120 cubic inches
  2. 240 cubic inches (correct answer)
  3. 480 cubic inches
  4. 1,200 cubic inches
Explanation: When you encounter prism volume problems, remember that the volume of any prism equals the area of its base multiplied by its height. This fundamental relationship applies whether you're dealing with rectangular, triangular, or other prismatic shapes. For this triangular prism, you're given that the triangular base has an area of 24 square inches and the prism's height is 10 inches. Using the volume formula: V=Base Area×Height=24×10=240 cubic inchesV = \text{Base Area} \times \text{Height} = 24 \times 10 = 240 \text{ cubic inches} Let's examine why the other answers are incorrect. Choice A (120 cubic inches) represents a common error where students might divide instead of multiply, or perhaps confuse this with a surface area calculation. Choice C (480 cubic inches) suggests doubling the correct answer, which might happen if you mistakenly think you need to account for both triangular faces of the prism. Choice D (1,200 cubic inches) is far too large and likely results from incorrectly applying a pyramid volume formula or making a calculation error with the given measurements. The key insight is recognizing that "height of the prism" refers to the perpendicular distance between the two triangular bases, not any measurement within the triangular base itself. The base area is already calculated for you at 24 square inches. For prism problems, always identify the base shape and its area first, then multiply by the prism's height. Don't overthink it—the volume formula for prisms is straightforward and consistent across all prism types.

Question 9

A cylindrical water tank is lying on its side. The tank has a radius of 4 feet and a length of 12 feet. What is the maximum volume of water it can hold?

  1. 96π cubic feet
  2. 144π cubic feet
  3. 192π cubic feet (correct answer)
  4. 576π cubic feet
Explanation: When you encounter a problem about the maximum volume a cylindrical tank can hold, you're dealing with the volume formula for a cylinder. The key insight here is that "maximum volume" means the tank is completely full, regardless of its orientation. To find the volume of a cylinder, use the formula V=πr2hV = \pi r^2 h, where rr is the radius and hh is the height (or in this case, the length). With a radius of 4 feet and length of 12 feet, you calculate: V=π×42×12=π×16×12=192πV = \pi \times 4^2 \times 12 = \pi \times 16 \times 12 = 192\pi cubic feet. Looking at the wrong answers: Choice A (96π) represents half the correct volume—you might get this if you mistakenly used radius instead of radius squared, calculating π×4×12\pi \times 4 \times 12. Choice B (144π) suggests you used 3 instead of 4 for the radius, giving π×32×16\pi \times 3^2 \times 16, or perhaps confused some of the given measurements. Choice D (576π) is three times the correct answer—this could result from using diameter instead of radius and making calculation errors, or from incorrectly applying the surface area formula. Remember that the phrase "lying on its side" is meant to distract you, but it doesn't affect the maximum volume calculation. Whether the cylinder is upright, on its side, or tilted, the total volume remains the same. Focus on identifying the radius and length correctly, then apply the volume formula systematically.

Question 10

A rectangular prism measures 4 cm by 3 cm by 5 cm; using V=lwhV=lwh, what is its volume?

  1. 60 cm360\text{ cm}^3 (correct answer)
  2. 12 cm312\text{ cm}^3
  3. 47 cm347\text{ cm}^3
  4. 94 cm394\text{ cm}^3
Explanation: This question tests ISEE Upper Level Mathematics Achievement, specifically the ability to calculate the volume of three-dimensional figures. Volume is the measure of space occupied by a 3D object, calculated by using specific formulas for different shapes. In this question, students must apply the volume formula V = l * w * h for a rectangular prism to calculate accurately. The correct answer, choice A, is derived by multiplying the dimensions 4 cm, 3 cm, and 5 cm, resulting in a volume of 60 cm³. Choice B is incorrect because it results from a common mistake of adding the dimensions instead of multiplying them. Teaching strategies include encouraging students to practice identifying the correct formula for each shape and performing step-by-step calculations to avoid errors. Emphasize checking unit consistency and understanding the relationship between volume and dimensions.

Question 11

A cube has a volume of 343 cubic inches. What is the length of each edge?

  1. 6 inches
  2. 7 inches (correct answer)
  3. 8 inches
  4. 9 inches
Explanation: When you encounter cube problems, remember that a cube has all edges of equal length, and volume equals edge length cubed: V=s3V = s^3, where ss is the side length. To find the edge length from a given volume of 343 cubic inches, you need to find the cube root of 343. This means asking: "What number, when multiplied by itself three times, gives 343?" Let's work backwards from the answer choices. For choice B) 7 inches: 73=7×7×7=49×7=3437^3 = 7 \times 7 \times 7 = 49 \times 7 = 343. This matches our given volume exactly, so 7 inches is correct. Let's verify why the other choices don't work: Choice A) 6 inches: 63=6×6×6=2166^3 = 6 \times 6 \times 6 = 216 cubic inches, which is too small. Choice C) 8 inches: 83=8×8×8=5128^3 = 8 \times 8 \times 8 = 512 cubic inches, which is too large. Choice D) 9 inches: 93=9×9×9=7299^3 = 9 \times 9 \times 9 = 729 cubic inches, which is much too large. The key insight is recognizing that 343 is a perfect cube. When you see volume problems with "nice" numbers like 343, the answer is likely a whole number, so testing the given choices by cubing them is often the fastest approach. Memorizing the first several perfect cubes (13=11^3 = 1, 23=82^3 = 8, 33=273^3 = 27, 43=644^3 = 64, 53=1255^3 = 125, 63=2166^3 = 216, 73=3437^3 = 343, etc.) will save you time on cube and cube root problems.

Question 12

A cone has a base radius of 5 inches and a volume of 100π cubic inches. What is the height of the cone?

  1. 4 inches
  2. 12 inches (correct answer)
  3. 20 inches
  4. 25 inches
Explanation: When you encounter cone volume problems, you're working with the formula V=13πr2hV = \frac{1}{3}\pi r^2 h, where V is volume, r is the base radius, and h is the height. The key is identifying which values you know and solving for the unknown. You're given that the base radius is 5 inches and the volume is 100π cubic inches. Substituting these values into the formula: 100π=13π(5)2h100\pi = \frac{1}{3}\pi (5)^2 h Simplifying: 100π=13π(25)h=25πh3100\pi = \frac{1}{3}\pi (25) h = \frac{25\pi h}{3} To solve for h, multiply both sides by 3 and divide by 25π: h=100π×325π=300π25π=12h = \frac{100\pi \times 3}{25\pi} = \frac{300\pi}{25\pi} = 12 The height is 12 inches, which is choice B. Looking at the wrong answers: Choice A (4 inches) would give you a volume of 13π(25)(4)=100π3\frac{1}{3}\pi(25)(4) = \frac{100\pi}{3}, which is too small. Choice C (20 inches) results from incorrectly using the full circle area formula (πr²) instead of the cone volume formula's 13\frac{1}{3} factor. Choice D (25 inches) comes from the common error of confusing the radius squared (25) with the height. Always write out the volume formula first when tackling cone problems, then carefully substitute your known values. Double-check that you're using 13πr2h\frac{1}{3}\pi r^2 h and not the cylinder formula, since forgetting the 13\frac{1}{3} factor is a frequent mistake on geometry problems.

Question 13

A cube has a surface area of 150 square feet. What is the volume of the cube?

  1. 125 cubic feet (correct answer)
  2. 216 cubic feet
  3. 343 cubic feet
  4. 729 cubic feet
Explanation: When you encounter cube problems, remember that all edges of a cube are equal, so if you know one measurement, you can find all others using the relationships between edge length, surface area, and volume. A cube has 6 identical square faces. If each edge has length ss, then each face has area s2s^2, making the total surface area 6s26s^2. Since the surface area is 150 square feet, you can write: 6s2=1506s^2 = 150. Solving for ss: s2=25s^2 = 25, so s=5s = 5 feet. Now that you know the edge length is 5 feet, you can find the volume using the formula V=s3V = s^3: V=53=125V = 5^3 = 125 cubic feet. Let's examine why the other answers are incorrect. Answer B (216 cubic feet) would result from an edge length of 6 feet, since 63=2166^3 = 216. However, a 6-foot edge would give a surface area of 6(62)=2166(6^2) = 216 square feet, not 150. Answer C (343 cubic feet) corresponds to 737^3, which would require an edge length of 7 feet and surface area of 294 square feet. Answer D (729 cubic feet) equals 939^3, implying a 9-foot edge and surface area of 486 square feet. The key strategy here is working systematically: surface area → edge length → volume. Don't try to jump directly from surface area to volume. Also, remember that cube problems often test whether you can move between different measurements of the same shape, so practice converting between edge length, surface area, and volume.

Question 14

A hemisphere has a radius of 9 centimeters. What is its volume?

  1. 243π cubic centimeters
  2. 486π cubic centimeters (correct answer)
  3. 729π cubic centimeters
  4. 972π cubic centimeters
Explanation: When you encounter hemisphere volume problems, remember that a hemisphere is exactly half of a sphere, so you'll need the sphere volume formula and then divide by 2. The volume of a sphere is V=43πr3V = \frac{4}{3}\pi r^3. For a hemisphere, this becomes V=12×43πr3=23πr3V = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3. With a radius of 9 centimeters, substitute into the hemisphere formula: V=23π(9)3=23π(729)=1458π3=486πV = \frac{2}{3}\pi (9)^3 = \frac{2}{3}\pi (729) = \frac{1458\pi}{3} = 486\pi cubic centimeters. This confirms that B) 486π cubic centimeters is correct. Looking at the wrong answers: A) 243π represents half of what you'd get if you incorrectly used r2r^2 instead of r3r^3 in your calculation. C) 729π would result from forgetting to apply the hemisphere factor entirely—this is actually 729π=(93)π729\pi = (9^3)\pi, suggesting you used just πr3\pi r^3 instead of the proper volume formula. D) 972π is what you'd get if you calculated the full sphere volume incorrectly as 43π(729)=972π\frac{4}{3}\pi (729) = 972\pi, then forgot to divide by 2 for the hemisphere. Study tip: Always write out both formulas when working with hemispheres: sphere volume =43πr3= \frac{4}{3}\pi r^3, then hemisphere volume =23πr3= \frac{2}{3}\pi r^3. This prevents the common mistake of forgetting the "half" factor or misremembering the original sphere formula.

Question 15

A right circular cone has a base area of 49π square centimeters and a volume of 196π cubic centimeters. What is the height of the cone?

  1. 4 centimeters
  2. 8 centimeters
  3. 12 centimeters (correct answer)
  4. 16 centimeters
Explanation: When you encounter cone volume problems, you need to connect three key measurements: base area, volume, and height using the cone volume formula. The volume of a cone is V=13×base area×heightV = \frac{1}{3} \times \text{base area} \times \text{height}. You're given that the base area is 49π49\pi square centimeters and the volume is 196π196\pi cubic centimeters. Substituting these values: 196π=13×49π×h196\pi = \frac{1}{3} \times 49\pi \times h To solve for height, first divide both sides by π\pi: 196=13×49×h196 = \frac{1}{3} \times 49 \times h Multiply both sides by 3: 588=49h588 = 49h Divide by 49: h=58849=12h = \frac{588}{49} = 12 centimeters This confirms answer C is correct. Let's examine why the other answers are wrong. Choice A (4 centimeters) would give a volume of 13×49π×4=196π3\frac{1}{3} \times 49\pi \times 4 = \frac{196\pi}{3}, which is too small. Choice B (8 centimeters) would yield 13×49π×8=392π3\frac{1}{3} \times 49\pi \times 8 = \frac{392\pi}{3}, still incorrect. Choice D (16 centimeters) would produce 13×49π×16=784π3\frac{1}{3} \times 49\pi \times 16 = \frac{784\pi}{3}, which is too large. These wrong answers likely result from computational errors or forgetting the 13\frac{1}{3} factor in the cone volume formula. Study tip: Always double-check your work by substituting your answer back into the original volume formula. This catches calculation mistakes and ensures you used the correct formula components.

Question 16

A spherical balloon has a radius of 6 inches. If the radius is increased by 50%, by what percent does the volume increase?

  1. 50%
  2. 125%
  3. 237.5% (correct answer)
  4. 337.5%
Explanation: When you encounter problems involving percentage changes in volume, remember that volume formulas contain the radius raised to a power, which amplifies the effect of any radius change. The volume of a sphere is V=43πr3V = \frac{4}{3}\pi r^3. With the original radius of 6 inches, the initial volume is V1=43π(6)3=43π(216)=288πV_1 = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (216) = 288\pi cubic inches. When the radius increases by 50%, the new radius becomes 6+0.5(6)=96 + 0.5(6) = 9 inches. The new volume is V2=43π(9)3=43π(729)=972πV_2 = \frac{4}{3}\pi (9)^3 = \frac{4}{3}\pi (729) = 972\pi cubic inches. The percent increase in volume is V2−V1V1×100%=972π−288π288π×100%=684π288π×100%=237.5%\frac{V_2 - V_1}{V_1} \times 100\% = \frac{972\pi - 288\pi}{288\pi} \times 100\% = \frac{684\pi}{288\pi} \times 100\% = 237.5\% Choice A (50%) incorrectly assumes the volume increases by the same percentage as the radius. Choice B (125%) might come from incorrectly thinking volume is proportional to r2r^2 instead of r3r^3, since (1.5)2=2.25(1.5)^2 = 2.25, giving a 125% increase. Choice D (337.5%) represents the ratio of new volume to old volume (972/288 = 3.375), but fails to subtract the original volume when calculating percent increase. Remember: when a linear dimension changes, areas change by the square of that factor, and volumes change by the cube. A 50% radius increase means the new radius is 1.5 times the original, so volume increases by (1.5)3=3.375(1.5)^3 = 3.375 times, representing a 237.5% increase.

Question 17

The volume of a cone is 84π cubic inches. If the radius is tripled and the height is halved, what is the new volume?

  1. 126π cubic inches
  2. 252π cubic inches
  3. 378π cubic inches (correct answer)
  4. 756π cubic inches
Explanation: When you encounter problems about changing dimensions of geometric shapes, you need to understand how each dimension affects the volume formula differently. The volume of a cone is V=13πr2hV = \frac{1}{3}\pi r^2 h. Starting with 84π cubic inches, let's see what happens when the radius is tripled and height is halved. If the original radius is rr and height is hh, then 13πr2h=84π\frac{1}{3}\pi r^2 h = 84\pi. With the new dimensions: radius becomes 3r3r and height becomes h2\frac{h}{2}. The new volume is: Vnew=13π(3r)2(h2)=13π⋅9r2⋅h2=96πr2h=32πr2hV_{new} = \frac{1}{3}\pi (3r)^2 \left(\frac{h}{2}\right) = \frac{1}{3}\pi \cdot 9r^2 \cdot \frac{h}{2} = \frac{9}{6}\pi r^2 h = \frac{3}{2}\pi r^2 h Since the original volume was 13πr2h=84π\frac{1}{3}\pi r^2 h = 84\pi, we can find πr2h=252π\pi r^2 h = 252\pi. Therefore: Vnew=32⋅252π=378πV_{new} = \frac{3}{2} \cdot 252\pi = 378\pi cubic inches. Choice A (126π) incorrectly assumes the volume increases by only 1.5 times, missing that radius is squared in the formula. Choice B (252π) represents 3×84π3 \times 84\pi, forgetting that tripling radius means multiplying by 32=93^2 = 9, not 3. Choice D (756π) correctly calculates 9×84π9 \times 84\pi for the tripled radius but forgets to divide by 2 for the halved height. Remember: when dimensions change in volume problems, radius affects volume quadratically (r2r^2), while height affects it linearly. Always substitute the new dimensions into the complete formula rather than trying mental shortcuts.

Question 18

A cube and a sphere have the same volume. If the cube has an edge length of 6 units, what is the radius of the sphere?

  1. 162π3\sqrt[3]{\frac{162}{π}} units (correct answer)
  2. 216π3\sqrt[3]{\frac{216}{π}} units
  3. 324π3\sqrt[3]{\frac{324}{π}} units
  4. 648π3\sqrt[3]{\frac{648}{π}} units
Explanation: When you encounter problems involving shapes with equal volumes, you need to set up equations using the volume formulas for each shape and solve for the unknown dimension. Start by finding the cube's volume. With an edge length of 6 units, the volume is 63=2166^3 = 216 cubic units. Since the sphere has the same volume, you can set up the equation: 43πr3=216\frac{4}{3}πr^3 = 216, where r is the sphere's radius. To solve for r, multiply both sides by 34π\frac{3}{4π}: r3=216×34π=6484π=162πr^3 = 216 × \frac{3}{4π} = \frac{648}{4π} = \frac{162}{π}. Taking the cube root gives you r=162π3r = \sqrt[3]{\frac{162}{π}} units. Looking at the wrong answers: Choice B gives 216π3\sqrt[3]{\frac{216}{π}}, which represents the error of forgetting to multiply by 34π\frac{3}{4π} and only dividing the cube's volume by π. Choice C shows 324π3\sqrt[3]{\frac{324}{π}}, which comes from incorrectly multiplying 216 by 32π\frac{3}{2π} instead of 34π\frac{3}{4π}. Choice D gives 648π3\sqrt[3]{\frac{648}{π}}, representing the mistake of multiplying by 3 but forgetting to divide by 4. The correct answer is A. Strategy tip: In equal volume problems, always write out both volume formulas completely, set them equal, and carefully track each step when isolating the unknown variable. The most common errors involve arithmetic mistakes when manipulating fractions with π.

Question 19

A composite solid consists of a cylinder topped with a hemisphere. The cylinder has a radius of 5 meters and height of 12 meters. What is the total volume?

  1. 300π+125π3300π + \frac{125π}{3} cubic meters
  2. 300π+250π3300π + \frac{250π}{3} cubic meters (correct answer)
  3. 400π+250π3400π + \frac{250π}{3} cubic meters
  4. 500π+500π3500π + \frac{500π}{3} cubic meters
Explanation: When you encounter composite solids, you need to find the volume of each individual shape and add them together. This problem combines a cylinder and a hemisphere, both sharing the same radius of 5 meters. For the cylinder, use the formula V=πr2hV = πr^2h. With radius 5 meters and height 12 meters: Vcylinder=π(5)2(12)=π(25)(12)=300πV_{cylinder} = π(5)^2(12) = π(25)(12) = 300π cubic meters. For the hemisphere, remember it's half of a sphere. The sphere volume formula is V=43πr3V = \frac{4}{3}πr^3, so a hemisphere is V=23πr3V = \frac{2}{3}πr^3. With radius 5 meters: Vhemisphere=23π(5)3=23π(125)=250π3V_{hemisphere} = \frac{2}{3}π(5)^3 = \frac{2}{3}π(125) = \frac{250π}{3} cubic meters. Total volume: 300π+250π3300π + \frac{250π}{3} cubic meters, which is answer choice B. Looking at the wrong answers: Choice A uses 125π3\frac{125π}{3} for the hemisphere volume, which would result from forgetting to multiply by 2 in the hemisphere formula—a common error when students use 13πr3\frac{1}{3}πr^3 instead of 23πr3\frac{2}{3}πr^3. Choice C incorrectly calculates the cylinder volume as 400π400π, possibly from using the wrong height. Choice D calculates the cylinder volume as 500π500π and uses 500π3\frac{500π}{3} for the hemisphere—both calculations contain significant computational errors. Remember: for composite solids, break down each component, apply the correct volume formulas, and carefully track your arithmetic. Double-check that you're using the hemisphere formula, not the full sphere formula.

Question 20

A cylinder has diameter 8 cm and height 10 cm; using V=πr2hV=\pi r^2h, what is its volume?

  1. 640π cm3640\pi\text{ cm}^3
  2. 160π cm3160\pi\text{ cm}^3 (correct answer)
  3. 320π cm3320\pi\text{ cm}^3
  4. 80π cm380\pi\text{ cm}^3
Explanation: This question tests ISEE Upper Level Mathematics Achievement, specifically the ability to calculate the volume of three-dimensional figures. Volume is the measure of space occupied by a 3D object, calculated by using specific formulas for different shapes. In this question, students must apply the volume formula V = π r² h for a cylinder to calculate accurately. The correct answer, choice B, is derived by halving the diameter to get radius 4 cm, squaring to 16, then multiplying by height 10 cm and π, resulting in 160π cm³. Choice A is incorrect because it results from a common mistake of using diameter instead of radius in squaring. Teaching strategies include encouraging students to practice identifying the correct formula for each shape and performing step-by-step calculations to avoid errors. Emphasize checking unit consistency and understanding the relationship between volume and dimensions.