Volume of Prisms - Pre Algebra

1. A rectangular storage unit is being designed with a square base. If the base has a side length of $s$ feet and the height is $(s + 4)$ feet, which expression represents the volume in cubic feet?
   Answer: $s^3 + 4s^2$ When you see a geometry problem asking for volume, you need to recall that volume measures the space inside a three-dimensional object. For any rectangular prism (including boxes with square bases), the volume formula is length × width × height. Since this storage unit has a square base with side length $s$ feet, both the length and width equal $s$. The height is given as $(s + 4)$ feet. Therefore, the volume is: $\text{Volume} = s \times s \times (s + 4) = s^2(s + 4)$ To expand this expression, you distribute $s^2$ to both terms inside the parentheses: $s^2(s + 4) = s^2 \cdot s + s^2 \cdot 4 = s^3 + 4s^2$ This confirms that choice C is correct. Looking at the wrong answers: Choice A ($s^3 + 4s$) incorrectly multiplies the height by $s$ instead of $s^2$, forgetting that the base is two-dimensional. Choice B ($s^2 + 4s$) represents the area of the base plus some linear term, but completely misses the three-dimensional nature of volume. Choice D ($2s^2 + 4s$) might come from incorrectly thinking about perimeter or surface area calculations. Remember this key strategy: always identify what type of measurement you're finding (length, area, or volume) and use the appropriate formula. For volume problems involving rectangular shapes, multiply all three dimensions together, then carefully expand any algebraic expressions using the distributive property.
2. A rectangular swimming pool has a length of 25 meters, a width of 15 meters, and an average depth of 2.4 meters. How many liters of water are needed to fill the pool? (1 cubic meter = 1000 liters)
   Answer: 900,000 liters This problem tests your ability to calculate volume and convert between units - two essential skills that often appear together on pre-algebra exams. To find how much water fills the pool, you need to calculate the pool's volume using the formula: Volume = length × width × depth. With dimensions of 25 meters × 15 meters × 2.4 meters, the volume is $25 \times 15 \times 2.4 = 900$ cubic meters. Since the question asks for liters, not cubic meters, you must convert using the given relationship: 1 cubic meter = 1000 liters. Therefore: $900 \text{ cubic meters} \times 1000 \text{ liters per cubic meter} = 900,000 \text{ liters}$. This confirms answer B is correct. Let's examine why the other options are wrong. Answer A (90,000 liters) represents forgetting to multiply by the depth - you'd get this if you only calculated $25 \times 15 \times 1000 = 375,000$, then made an additional error. Answer C (900 liters) is the volume in cubic meters without converting to liters - a classic unit conversion mistake. Answer D (9,000 liters) suggests multiplying the volume by 10 instead of 1000, showing confusion about the conversion factor. Study tip: When solving volume problems, always work in two clear steps: first calculate the volume using length × width × height, then convert units if necessary. Write down your conversion factor (like 1 m³ = 1000 L) to avoid mixing up whether to multiply or divide.
3. Two identical rectangular prisms are placed end-to-end along their length to form a larger rectangular prism. If each original prism has dimensions 4 cm × 6 cm × 10 cm, what is the volume of the combined prism?
   Answer: 480 cubic cm Each original prism has volume = 4 × 6 × 10 = 240 cubic cm. When two identical prisms are combined, the total volume is simply twice the volume of one prism: 2 × 240 = 480 cubic cm. The combined prism has dimensions 4 cm × 6 cm × 20 cm (length doubled), giving volume = 4 × 6 × 20 = 480 cubic cm. Choice B is the volume of just one prism. Choice C incorrectly multiplies by 4 instead of 2. Choice D incorrectly multiplies by 3.
4. A warehouse has three identical storage rooms, each in the shape of a rectangular prism. If one room has a volume of 2,400 cubic feet and dimensions of 20 feet by 8 feet by $h$ feet, what is the total volume of all three storage rooms?
   Answer: 7,200 cubic feet This question combines two key concepts: finding the volume of a rectangular prism and understanding what "total" means when dealing with multiple identical objects. First, you need to find the height of one storage room using the volume formula for rectangular prisms: $V = l \times w \times h$. Since one room has a volume of 2,400 cubic feet with dimensions 20 feet by 8 feet by $h$ feet, you can write: $2,400 = 20 \times 8 \times h$. Simplifying: $2,400 = 160h$, so $h = 15$ feet. Now that you know one room has a volume of 2,400 cubic feet, and there are three identical rooms, the total volume is $3 \times 2,400 = 7,200$ cubic feet. Looking at the wrong answers: Choice A (2,400 cubic feet) gives you only the volume of one room—you forgot to multiply by three. Choice C (4,800 cubic feet) represents two rooms instead of three, suggesting you miscounted. Choice D (3,600 cubic feet) might result from calculation errors, possibly multiplying the room's dimensions incorrectly or making an arithmetic mistake. The key strategy here is to read carefully and identify what the question is actually asking for. When you see "total volume of all three storage rooms," make sure your final step multiplies the individual volume by the number of rooms. Many students solve for one room's volume correctly but forget this final multiplication step.
5. A rectangular fish tank is filled with water to a height of 8 inches. The tank has a base area of 240 square inches. If 960 cubic inches of water are removed, what is the new height of the water?
   Answer: 4 inches This problem tests your understanding of volume and how changes in volume affect the dimensions of rectangular prisms. When you see questions about removing liquid from containers, think about how volume, base area, and height relate to each other. First, find the original volume of water using the formula $\text{Volume} = \text{Base Area} \times \text{Height}$. With a base area of 240 square inches and height of 8 inches, the original volume is $240 \times 8 = 1,920$ cubic inches. After removing 960 cubic inches, the remaining volume is $1,920 - 960 = 960$ cubic inches. Since the base area stays the same (240 square inches), you can find the new height: $\text{New Height} = \frac{\text{New Volume}}{\text{Base Area}} = \frac{960}{240} = 4$ inches. Looking at the wrong answers: Choice A (3 inches) would give a remaining volume of only 720 cubic inches, meaning too much water was removed. Choice B (6 inches) incorrectly assumes you subtract 2 inches from the original height (8 - 2 = 6), but this ignores the actual volume calculation. Choice C (5 inches) would leave 1,200 cubic inches of water, which means only 720 cubic inches were removed instead of 960. The key strategy here is to always work with volumes first when liquid is added or removed, then convert back to the dimension you need. Don't try to directly calculate height changes—let the volume relationship guide you to the correct answer.
6. A concrete foundation is poured in the shape of a rectangular prism. The foundation is 20 feet long, 12 feet wide, and 1.5 feet deep. If concrete costs $8.50 per cubic foot, what is the total cost of the concrete needed?
   Answer: $3,060 When you encounter a word problem involving the cost of materials for a 3D shape, you need to find the volume first, then multiply by the unit price. This is a two-step process that combines geometry with basic arithmetic. To find the volume of a rectangular prism (like this foundation), you multiply length × width × height. Here, that's $20 \text{ ft} \times 12 \text{ ft} \times 1.5 \text{ ft} = 360 \text{ cubic feet}$. Next, multiply the volume by the cost per cubic foot: $360 \times \$8.50 = \$3,060$. Looking at the wrong answers: Choice A ($2,448) likely comes from miscalculating the volume—perhaps using 20 × 12 × 1.44 instead of 1.5, or making an arithmetic error in the multiplication. Choice B ($360) is just the volume in cubic feet, not the total cost—this happens when you forget the second step of multiplying by the price per cubic foot. Choice D ($306) appears to come from a decimal error, possibly calculating 360 × $0.85 instead of 360 × $8.50, essentially moving the decimal point in the wrong direction. The correct answer is C ($3,060). Study tip: For any "cost of materials" problem involving 3D shapes, always work in two clear steps: (1) find the volume using the appropriate formula, and (2) multiply volume by unit cost. Write both steps down separately to avoid skipping the cost calculation or making decimal errors.

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