Rectangular Prism Volume - SSAT Middle Level Quantitative

A rectangular fish tank is 60 cm long, 30 cm wide, and 40 cm tall. If the water level is currently 25 cm high, how many more liters of water are needed to fill the tank completely? (Note: 1 liter = 1,000 cubic cm)
Answer: 27 liters This problem tests your understanding of volume calculations and unit conversions. When you see a question about filling containers, you need to find the difference between the total capacity and the current amount. First, calculate the tank's total volume: $60 \times 30 \times 40 = 72,000 \text{ cubic cm}$ . Next, find the current water volume using the actual water height of 25 cm: $60 \times 30 \times 25 = 45,000 \text{ cubic cm}$ . The additional water needed is $72,000 - 45,000 = 27,000 \text{ cubic cm}$ . Converting to liters: $27,000 ÷ 1,000 = 27 \text{ liters}$ , which is answer A. Looking at the wrong answers: B) 45 liters represents the current volume of water already in the tank, not the additional amount needed. C) 54 liters might result from miscalculating the empty space (perhaps using incorrect dimensions or making an arithmetic error). D) 72 liters is the tank's total capacity, but the question asks for additional water needed, not total capacity. The key trap here is confusing what the question asks for. Many students calculate either the current water volume or total tank capacity instead of the difference between them. Always read carefully to distinguish between "how much is there," "how much fits total," and "how much more is needed." Practice breaking volume problems into clear steps: find total capacity, find current amount, then subtract to find the difference.
Three identical rectangular boxes are stacked on top of each other. Each box has dimensions 10 cm by 8 cm by 6 cm. What is the total volume of all three boxes?
Answer: 1,440 cubic centimeters When you encounter questions about stacking identical objects, remember that stacking doesn't change the individual volumes - you're simply adding up the volumes of separate objects. To find the volume of a rectangular box, you multiply length × width × height. For each box with dimensions 10 cm by 8 cm by 6 cm, the volume is $10 \times 8 \times 6 = 480$ cubic centimeters. Since you have three identical boxes stacked together, the total volume is simply three times the volume of one box: $3 \times 480 = 1,440$ cubic centimeters. This makes C the correct answer. Let's examine why the other options are wrong. Choice A (480 cubic centimeters) represents the volume of just one box - this would be the answer if you forgot to multiply by three. Choice B (960 cubic centimeters) equals $2 \times 480$ , which would be correct for only two boxes, not three. Choice D (1,920 cubic centimeters) equals $4 \times 480$ , representing four boxes instead of three. The key strategy here is to break multi-step volume problems into clear parts: first calculate the volume of one unit, then multiply by the number of units. Don't let the word "stacked" confuse you - stacking identical objects is just addition of volumes. Always double-check that you're multiplying by the correct number of objects mentioned in the problem.
A shipping container in the shape of a rectangular prism has interior dimensions of 20 feet by 8 feet by 8.5 feet. If the container is loaded with identical boxes that each measure 2 feet by 2 feet by 1.5 feet, and the boxes must be placed with their largest face down, what is the maximum number of boxes that can fit?
Answer: 200 boxes When you encounter a packing problem like this, you need to think systematically about how objects fit within a container's constraints. The key insight here is understanding what "largest face down" means and how to maximize the arrangement. First, identify the largest face of each box. The boxes measure 2 feet by 2 feet by 1.5 feet, so the largest face is 2×2 feet (area = 4 square feet). With this face down, each box has a height of 1.5 feet. Now calculate how many boxes fit in each dimension of the container (20×8×8.5 feet): - Length: $20 ÷ 2 = 10$ boxes - Width: $8 ÷ 2 = 4$ boxes - Height: $8.5 ÷ 1.5 = 5.67$ , so 5 complete layers Total boxes: $10 × 4 × 5 = 200$ boxes. Choice A (136 boxes) likely comes from miscalculating the floor dimensions or making an error with the height calculation. Choice B (160 boxes) suggests calculating only 4 layers instead of 5, perhaps by incorrectly thinking 8.5÷1.5 = 4. Choice C (180 boxes) might result from using 4.5 layers and rounding incorrectly, but you can't have partial boxes. The correct answer is D (200 boxes). Study tip: In packing problems, always divide each container dimension by the corresponding box dimension and take the floor of the result (since you can't have partial boxes). Then multiply all three results together. Pay special attention to orientation constraints like "largest face down."
A rectangular concrete foundation has dimensions 30 meters by 18 meters by 0.5 meters. If concrete costs $120 per cubic meter, what is the total cost of the concrete needed?
Answer: $32,400 This is a volume and cost calculation problem that tests your ability to find the volume of a rectangular prism and apply unit rates. When you see dimensions given for a three-dimensional object along with a cost per unit volume, you need to calculate the total volume first, then multiply by the unit cost. To find the volume of the rectangular foundation, multiply length × width × height: $30 \text{ m} \times 18 \text{ m} \times 0.5 \text{ m} = 270 \text{ cubic meters}$ . Then multiply the volume by the cost per cubic meter: $270 \times \$120 = \$32,400$ . Looking at why each answer choice is wrong: Choice B ($28,800) likely comes from miscalculating the volume as 240 cubic meters instead of 270, perhaps by making an arithmetic error in the multiplication. Choice C ($25,200) suggests a volume calculation of 210 cubic meters, which could result from incorrectly using 28 meters instead of 30 meters for one dimension. Choice D ($21,600) corresponds to a volume of 180 cubic meters, possibly from using 20 meters instead of 30 meters for the length, or making another significant calculation error in the volume step. The correct answer is A ($32,400). Strategy tip: For volume-cost problems, always work in two clear steps: calculate volume first, then multiply by unit cost. Double-check your volume calculation by ensuring you've used all three dimensions correctly, and be especially careful with decimal measurements like the 0.5 meters here.
A rectangular box has dimensions where the length is 3 times the width, and the height equals the width. If the width is 5 inches, what is the volume of the box?
Answer: 375 cubic inches When you encounter a volume problem with given relationships between dimensions, start by identifying each dimension clearly, then apply the volume formula for a rectangular box: $V = \text{length} \times \text{width} \times \text{height}$ . Given that the width is 5 inches, you can find the other dimensions using the stated relationships. Since the length is 3 times the width, the length equals $3 \times 5 = 15$ inches. Since the height equals the width, the height is 5 inches. Now calculate the volume: $V = 15 \times 5 \times 5 = 375$ cubic inches. This confirms answer choice C is correct. Let's examine why the other options are wrong. Choice A (325 cubic inches) would result from incorrectly calculating $13 \times 5 \times 5$ , possibly from misreading "3 times the width" as "3 more than the width." Choice B (350 cubic inches) might come from using $14 \times 5 \times 5$ , another misinterpretation of the length relationship. Choice D (400 cubic inches) would result from calculating $16 \times 5 \times 5$ , perhaps from adding instead of multiplying when finding the length. Strategy tip: Always write out the relationships as equations first. If width = w, then length = 3w and height = w. This prevents confusion between "times" (multiplication) and "more than" (addition). Double-check that your final answer has the correct units—volume should always be in cubic units.
Three rectangular boxes are placed end-to-end to form a longer rectangular shape. Each box measures 8 inches by 6 inches by 10 inches. If they are arranged with their 8-inch sides touching, what is the volume of the combined shape?
Answer: 1,440 cubic inches When you encounter problems involving multiple objects arranged together, the key is understanding how the arrangement affects the overall dimensions while keeping volume relationships clear. First, let's find the volume of one box: $8 \times 6 \times 10 = 480$ cubic inches. Since you have three identical boxes, the total volume will be $3 \times 480 = 1,440$ cubic inches, regardless of how they're arranged. This is a crucial insight: the arrangement changes the shape's external dimensions but never changes the total volume. The phrase "arranged with their 8-inch sides touching" describes how the boxes are positioned but doesn't affect the volume calculation. If you were asked for the combined shape's dimensions, this detail would matter (the new shape would be 24 inches by 6 inches by 10 inches), but for volume, you simply need the sum of individual volumes. Looking at the wrong answers: B) 1,680 might result from incorrectly calculating one box's volume as 280 cubic inches, then multiplying by 6. C) 1,920 could come from miscalculating a single box as 640 cubic inches, or from somehow using 4 boxes instead of 3. D) 2,160 suggests calculating one box as 720 cubic inches, possibly by incorrectly doubling one dimension. Strategy tip: For combination problems, calculate individual volumes first, then add them up. The physical arrangement only matters if you're asked about surface area or external dimensions, not total volume. Always verify your single-unit calculation before multiplying.
A rectangular aquarium has dimensions of 12 inches by 8 inches by 10 inches. How many cubic inches of water are needed to fill the aquarium to exactly 75% of its capacity?
Answer: 720 cubic inches This problem tests your understanding of volume calculations and percentage applications. When you see questions involving rectangular containers and partial filling, you need to find the total volume first, then calculate the specified percentage. To find how much water fills 75% of the aquarium, start by calculating the total volume using the formula: length × width × height. The aquarium's volume is $12 \times 8 \times 10 = 960$ cubic inches. Since you need 75% of this capacity, multiply the total volume by 0.75: $960 \times 0.75 = 720$ cubic inches. This confirms that choice A is correct. Now let's examine why the other answers are wrong. Choice B (900 cubic inches) likely comes from miscalculating 75% as $960 \times 0.9375 = 900$ , which would actually be 93.75% of the capacity. Choice C (960 cubic inches) represents the total volume of the aquarium at 100% capacity, not the 75% requested. Choice D (1200 cubic inches) appears to come from incorrectly calculating the dimensions, perhaps using $12 \times 10 \times 10 = 1200$ instead of the correct measurements. Remember this two-step approach for percentage volume problems: first calculate the total volume of the container, then multiply by the decimal form of the percentage. Also, always double-check that your final answer is smaller than the total volume when dealing with partial capacity questions.
A rectangular prism has a length of 15 meters, a width of 8 meters, and a volume of 840 cubic meters. What is the height of the prism?
Answer: 7 meters When you encounter volume problems with rectangular prisms, remember that volume equals length times width times height: $V = l \times w \times h$ . Since you're given three of these four values, you can solve for the missing dimension. You know the volume is 840 cubic meters, length is 15 meters, and width is 8 meters. Substitute these values into the formula: $840 = 15 \times 8 \times h$ . First, multiply the known dimensions: $15 \times 8 = 120$ . So your equation becomes $840 = 120 \times h$ . To find the height, divide both sides by 120: $h = 840 \div 120 = 7$ meters. Let's verify why the other answers don't work. Choice (A) 6 meters would give you a volume of $15 \times 8 \times 6 = 720$ cubic meters, which is 120 cubic meters too small. Choice (C) 8 meters would yield $15 \times 8 \times 8 = 960$ cubic meters, exceeding the given volume by 120 cubic meters. Choice (D) 9 meters would produce $15 \times 8 \times 9 = 1,080$ cubic meters, far too large at 240 cubic meters over the target. The correct answer is (B) 7 meters. Study tip: Always double-check volume problems by multiplying your answer back with the given dimensions. This catches calculation errors quickly. Also, when one dimension equals the width (like choice C), it's often a distractor designed to tempt students who confuse the given measurements.
Two rectangular boxes have the same volume of 240 cubic centimeters. Box A has dimensions 6 cm by 5 cm by 8 cm. If Box B has a length of 10 cm and a width of 4 cm, what is the height of Box B?
Answer: 6 cm When you encounter volume problems with rectangular boxes, remember that volume equals length × width × height. The key insight here is that if two boxes have the same volume, you can use this relationship to find missing dimensions. First, let's verify Box A's volume: $6 \times 5 \times 8 = 240$ cubic centimeters. This confirms the given volume. For Box B, we know the volume must also be 240 cubic centimeters, the length is 10 cm, and the width is 4 cm. Using the volume formula: $10 \times 4 \times h = 240$ , where h is the unknown height. Simplifying: $40h = 240$ , so $h = 6$ cm. Looking at the wrong answers: Choice A (4 cm) would give Box B a volume of $10 \times 4 \times 4 = 160$ cubic centimeters, which is too small. Choice C (8 cm) would result in $10 \times 4 \times 8 = 320$ cubic centimeters, exceeding the required volume. Choice D (10 cm) would create $10 \times 4 \times 10 = 400$ cubic centimeters, which is significantly too large. These incorrect choices might tempt you if you accidentally use dimensions from Box A or make arithmetic errors. Choice B (6 cm) gives us $10 \times 4 \times 6 = 240$ cubic centimeters, matching the required volume perfectly. Study tip: Always double-check your volume calculations by multiplying your answer back into the formula. This catches arithmetic mistakes and ensures your solution makes sense within the problem's constraints.
Two rectangular prisms have the same height of 5 cm. The first has a base of 6 cm by 8 cm, and the second has a square base. If both prisms have the same volume, what is the side length of the square base?
Answer: 6.9 cm When you encounter problems comparing volumes of different shaped prisms, remember that volume equals base area times height. Since both prisms have the same height and volume, their base areas must be equal. First, find the volume of the first prism. With a base of 6 cm × 8 cm and height of 5 cm, its volume is $6 \times 8 \times 5 = 240 \text{ cm}^3$ . Since both prisms have the same volume, the second prism also has a volume of 240 cm³. For the second prism with a square base and height of 5 cm, if the side length is $s$ , then: $s^2 \times 5 = 240$ . Solving for $s$ : $s^2 = 48$ , so $s = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \approx 6.9 \text{ cm}$ . Looking at the wrong answers: Choice A (6.0 cm) gives a volume of $6^2 \times 5 = 180 \text{ cm}^3$ , which is too small. Choice C (7.2 cm) yields $7.2^2 \times 5 = 259.2 \text{ cm}^3$ , which exceeds our target. Choice D (7.8 cm) produces $7.8^2 \times 5 = 304.2 \text{ cm}^3$ , which is significantly too large. Only choice B (6.9 cm) gives approximately the correct volume of 240 cm³. Study tip: When comparing volumes of different shapes with equal volumes, set up equations using the volume formulas and solve algebraically. Always check your answer by substituting back into the original volume formula.
A rectangular room has a volume of 2,400 cubic feet. If the length is 20 feet and the width is 15 feet, and the ceiling height is increased by 25%, what will be the new volume?
Answer: 3,000 cubic feet When you encounter volume problems involving rectangular shapes, remember that volume equals length × width × height. The key here is understanding how percentage changes affect the final result. First, find the original height using the given volume. Since $\text{Volume} = \text{length} \times \text{width} \times \text{height}$ , you have $2400 = 20 \times 15 \times h$ . This gives you $2400 = 300h$ , so $h = 8$ feet. Next, calculate the new height after a 25% increase. A 25% increase means multiplying by 1.25: $8 \times 1.25 = 10$ feet. Finally, find the new volume with the increased height: $20 \times 15 \times 10 = 3000$ cubic feet. The answer is B. Looking at the wrong choices: A) 3,200 represents adding 25% to the original volume ( $2400 + 0.25 \times 2400 = 3200$ ), but this incorrectly assumes the volume increases by 25% when only the height does. C) 2,700 might result from miscalculating the height increase as 20% instead of 25%. D) 3,600 could come from incorrectly thinking a 25% height increase means a 50% volume increase. Remember: when one dimension of a rectangular solid changes by a percentage, the volume changes by that same percentage only if the other dimensions stay constant. Don't confuse changes in individual dimensions with changes in the total volume.
A rectangular water tank has a volume of 7,200 cubic inches. The tank is 20 inches long and 18 inches wide. If water is added at a rate of 150 cubic inches per minute, how long will it take to fill the tank completely?
Answer: 48 minutes When you encounter a problem involving rates and volumes, you need to find the total capacity first, then determine how long it takes to fill at the given rate. To find the tank's height, use the volume formula for rectangular prisms: $\text{Volume} = \text{length} \times \text{width} \times \text{height}$ . Since you know the volume is 7,200 cubic inches, length is 20 inches, and width is 18 inches, you can solve: $7,200 = 20 \times 18 \times h$ . This gives you $7,200 = 360h$ , so $h = 20$ inches. Now that you've confirmed the tank's dimensions make sense, use the rate formula: $\text{Time} = \frac{\text{Total Volume}}{\text{Rate}}$ . With a volume of 7,200 cubic inches and a filling rate of 150 cubic inches per minute: $\text{Time} = \frac{7,200}{150} = 48$ minutes. Choice A (45 minutes) likely results from miscalculating the division or rounding incorrectly. Choice C (52 minutes) might come from using 138.46 as the rate (7,200 ÷ 52) and working backwards incorrectly. Choice D (56 minutes) could result from calculation errors or misreading the given dimensions. The correct answer is B: 48 minutes. Strategy tip: For rate problems, always identify what you're solving for first. If you need total capacity but only have some dimensions, use the volume formula to find missing measurements. Then apply the rate formula systematically. Double-check your arithmetic, as these problems often involve straightforward division that's easy to miscalculate under time pressure.
A rectangular tank has a base area of 48 square feet and a height of 7 feet. If water is added until the tank is 3/4 full, how many cubic feet of water are in the tank?
Answer: 252 cubic feet Volume problems involving partially filled containers require you to find the total capacity first, then calculate the portion that's actually filled. To find the volume of water, you need to determine the tank's total volume and then find three-fourths of that amount. The volume of a rectangular tank equals base area × height, so the total capacity is $48 \times 7 = 336$ cubic feet. Since the tank is $\frac{3}{4}$ full, you multiply the total volume by this fraction: $336 \times \frac{3}{4} = \frac{1008}{4} = 252$ cubic feet of water. Looking at the wrong answers: Choice B (288 cubic feet) likely comes from incorrectly calculating $48 \times 6$ instead of properly finding three-fourths of the total volume. Choice C (324 cubic feet) might result from miscalculating the fraction, perhaps finding $\frac{27}{28}$ of the total volume instead of $\frac{3}{4}$ . Choice D (336 cubic feet) is the total capacity of the tank when completely full, not when it's three-fourths full. The correct answer is A: 252 cubic feet. When tackling volume problems with partial filling, always work in two clear steps: first find the container's total capacity, then multiply by the fraction that indicates how full it is. Don't try to shortcut by multiplying the height by the fraction first—this approach often leads to calculation errors and matches common wrong answer choices.
A rectangular box has dimensions in the ratio 3:4:5. If the smallest dimension is 6 inches, what is the volume of the box?
Answer: 480 cubic inches When you encounter ratio problems involving 3D shapes, you need to use the given ratio to find all dimensions, then calculate volume using the standard formula. Since the dimensions are in the ratio 3:4:5 and the smallest dimension is 6 inches, you can set up the problem systematically. The smallest ratio number is 3, so if 3 corresponds to 6 inches, then each ratio unit equals $6 ÷ 3 = 2$ inches. This means the three dimensions are: $3 × 2 = 6$ inches, $4 × 2 = 8$ inches, and $5 × 2 = 10$ inches. The volume of a rectangular box is length × width × height, so: $6 × 8 × 10 = 480$ cubic inches. This confirms that A is correct. Let's examine why the other answers are wrong. Answer B (720 cubic inches) might result from incorrectly using 3 as a multiplier instead of finding the unit value, leading to dimensions like 9, 12, and 15. Answer C (960 cubic inches) could come from doubling the correct volume or making an arithmetic error in the multiplication. Answer D (1,200 cubic inches) might result from using the ratio numbers directly without proper scaling, perhaps calculating $3 × 4 × 5 × 20 = 1,200$ . Remember this strategy: in ratio problems, always identify which ratio part corresponds to the given measurement, find the unit value by dividing, then multiply each ratio part by this unit value. Double-check by ensuring your calculated dimensions maintain the original ratio.
A storage shed has interior dimensions of 12 feet by 9 feet by 8 feet. If boxes measuring 3 feet by 3 feet by 2 feet are placed inside, what is the maximum number of boxes that can fit?
Answer: 48 boxes When you encounter a problem about fitting boxes into a storage space, you're dealing with three-dimensional packing. The key insight is determining how many boxes fit along each dimension of the storage shed. To find the maximum number of boxes, divide each dimension of the shed by the corresponding dimension of the boxes. The shed measures 12 feet by 9 feet by 8 feet, and each box measures 3 feet by 3 feet by 2 feet. Along the 12-foot dimension: $12 ÷ 3 = 4$ boxes Along the 9-foot dimension: $9 ÷ 3 = 3$ boxes Along the 8-foot dimension: $8 ÷ 2 = 4$ boxes The total number of boxes is $4 × 3 × 4 = 48$ boxes. Choice A (24 boxes) represents what you'd get if you miscalculated one dimension, perhaps thinking only 2 boxes fit along the 8-foot height instead of 4. Choice B (32 boxes) might result from incorrectly assuming the boxes must be oriented differently or making an arithmetic error in the multiplication. Choice C (36 boxes) could come from multiplying $4 × 3 × 3$ , which would happen if you mistakenly thought only 3 boxes fit along the 8-foot dimension instead of 4. Remember that in packing problems, you can orient boxes to maximize fit as long as the dimensions work. Always check that each box dimension fits properly into the corresponding storage dimension, then multiply the number of boxes that fit along each axis. This systematic approach prevents calculation errors.
A rectangular room is 16 feet long, 12 feet wide, and 9 feet high. If the room is divided into two equal parts by a wall parallel to the width, what is the volume of each smaller room?
Answer: 864 cubic feet When you encounter volume problems involving room divisions, remember that volume equals length × width × height, and you need to determine how the division affects these dimensions. The original room has dimensions 16 feet long, 12 feet wide, and 9 feet high, giving it a total volume of $16 \times 12 \times 9 = 1,728$ cubic feet. The key insight is understanding what "divided by a wall parallel to the width" means. Since the wall is parallel to the width (12 feet), it runs across the length dimension, splitting the 16-foot length into two 8-foot sections. Each smaller room now measures 8 feet long, 12 feet wide, and 9 feet high. The volume of each smaller room is $8 \times 12 \times 9 = 864$ cubic feet, confirming that A is correct. Looking at the wrong answers: B (1,152) might result from incorrectly dividing the width instead of length, creating rooms that are 16 × 6 × 9. C (1,296) could come from mistakenly dividing the height, yielding 16 × 12 × 4.5, then rounding errors. D (1,728) is the original room's total volume—a trap for students who forget to account for the division entirely. Remember this pattern: when a room is divided by a wall "parallel to" a dimension, that wall runs along that dimension but splits the perpendicular dimension in half. Always identify which measurement gets divided, then calculate the new volume using the modified dimensions.
A rectangular swimming pool has a length that is twice its width. If the width is 12 feet, the depth is 5 feet, and the pool is filled to 80% capacity, how many cubic feet of water are in the pool?
Answer: 1,152 cubic feet This problem tests your understanding of volume calculations and percentages. When you see a rectangular pool problem, you need to find the volume using length × width × depth, then apply any additional conditions. First, let's find the pool's dimensions. The width is given as 12 feet, and the length is twice the width, so the length is $2 \times 12 = 24$ feet. The depth is 5 feet. The total volume of the pool would be $24 \times 12 \times 5 = 1,440$ cubic feet. However, the pool is only filled to 80% capacity, so the actual volume of water is $1,440 \times 0.80 = 1,152$ cubic feet. Looking at the wrong answers: Choice A (960) represents a common error where students might miscalculate the dimensions or forget to properly apply the 80% factor. Choice C (1,200) could result from incorrectly calculating 80% or making an arithmetic error in the volume calculation. Choice D (1,440) is the trap answer that gives you the total volume of the pool without accounting for the 80% fill level—this is what many students calculate first and mistakenly select. The correct answer is B: 1,152 cubic feet. Study tip: In volume problems involving percentages, always calculate the full volume first, then apply the percentage. Write out each step clearly to avoid the common trap of forgetting to apply percentage conditions. Watch for keywords like "filled to capacity" that modify your final calculation.
A rectangular swimming pool has a constant depth of 4 feet. The pool is 25 feet long and contains 2,000 cubic feet of water. What is the width of the pool?
Answer: 20 feet When you encounter a problem involving the volume of a rectangular pool, you're working with the formula for the volume of a rectangular prism: $\text{Volume} = \text{length} \times \text{width} \times \text{height (depth)}$ . Given that the pool contains 2,000 cubic feet of water, is 25 feet long, and has a depth of 4 feet, you can substitute these values into the formula: $2,000 = 25 \times \text{width} \times 4$ . Simplifying the right side: $2,000 = 100 \times \text{width}$ . Solving for width: $\text{width} = \frac{2,000}{100} = 20 \text{ feet}$ . This confirms that choice B is correct. Let's examine why the other options are incorrect. Choice A (18 feet) would give a volume of $25 \times 18 \times 4 = 1,800$ cubic feet, which is 200 cubic feet too small. Choice C (22 feet) would result in $25 \times 22 \times 4 = 2,200$ cubic feet, which exceeds the given volume by 200 cubic feet. Choice D (24 feet) would produce $25 \times 24 \times 4 = 2,400$ cubic feet, overshooting by 400 cubic feet. For volume problems on the SSAT, always identify what you know and what you need to find, then substitute carefully into the appropriate formula. Double-check your arithmetic by plugging your answer back into the original equation—this catches calculation errors and builds confidence in your solution.
A cube has the same volume as a rectangular prism with dimensions 4 cm by 6 cm by 9 cm. What is the length of one side of the cube?
Answer: 6 cm When you encounter problems comparing volumes of different shapes, you need to find the volumes of both shapes and set them equal to solve for the unknown dimension. First, find the volume of the rectangular prism using the formula $V = length \times width \times height$ : $V = 4 \times 6 \times 9 = 216 \text{ cm}^3$ Since the cube has the same volume, its volume is also 216 cm³. For a cube, all sides are equal, so if each side length is $s$ , then $V = s^3$ . Setting up the equation: $s^3 = 216$ To find $s$ , you need the cube root of 216. You can work this out by testing perfect cubes or factoring: $216 = 6^3$ , so $s = 6$ cm. Looking at the wrong answers: Choice B (8 cm) would give a volume of $8^3 = 512 \text{ cm}^3$ , which is much too large. Choice C (9 cm) might tempt you because 9 is one of the rectangular prism's dimensions, but $9^3 = 729 \text{ cm}^3$ is far too big. Choice D (12 cm) gives $12^3 = 1728 \text{ cm}^3$ , which is enormous compared to our target volume. The answer is A. Study tip: When comparing volumes of different shapes, always calculate the known volume first, then use the appropriate formula for the unknown shape. Don't assume that matching dimensions between shapes means anything—volume calculations require using the complete formulas for each shape type.
A warehouse has interior dimensions of 40 meters by 25 meters by 12 meters. If it is filled to 85% of its capacity with goods, how many cubic meters of empty space remain?
Answer: 1,800 cubic meters When you encounter a problem about "remaining space" or "empty space," you're dealing with a volume calculation that requires finding what's left after a portion is used. Start by calculating the total volume of the warehouse: $40 \times 25 \times 12 = 12,000$ cubic meters. Since the warehouse is filled to 85% capacity, the goods occupy $12,000 \times 0.85 = 10,200$ cubic meters. The empty space remaining is the total volume minus the occupied space: $12,000 - 10,200 = 1,800$ cubic meters. Looking at the wrong answers: Choice B (1,500 cubic meters) likely comes from miscalculating the percentage—perhaps using 87.5% instead of 85%, which would leave 1,500 cubic meters empty. Choice C (1,200 cubic meters) represents exactly 10% of the total volume, suggesting someone might have confused this with a different percentage calculation. Choice D (1,000 cubic meters) could result from calculation errors in either the volume computation or percentage work. The correct answer is A) 1,800 cubic meters. For problems involving percentages of volume, always work systematically: find total volume first, then calculate what portion is used, and finally subtract to find what remains. Watch out for percentage confusion—make sure you're calculating the right portion (filled vs. empty) and double-check your decimal conversions when working with percentages.

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