Chapter 3: Problem 57
A large cube has edges that are twice as long as those of a small cube. The volume of the large cube is how many times the volume of the small cube? A. 2 A. 4 C. 6 D. 8 E. 16
Short Answer
Expert verified
A. 2
B. 4
C. 6
D. 8
Answer: D. 8
Step by step solution
01
Express the large cube's edge length in terms of the small cube's edge length
We are given that the edges of the large cube are twice as long as those of the small cube. If the small cube has an edge length of x, then the large cube has an edge length of 2x.
02
Find the volume of the small cube
The volume of a cube is given by the formula V = s^3, where s is the edge length. In this case, the edge length of the small cube is x, so its volume is V_s = x^3.
03
Find the volume of the large cube
Using the same formula as in Step 2, but with the edge length of the large cube (2x), we find the volume of the large cube: V_L = (2x)^3.
04
Calculate (2x)^3
To calculate (2x)^3, we follow the exponentiation rules, which state that (a*b)^c = a^c * b^c. Therefore, (2x)^3 = 2^3 * x^3 = 8 * x^3.
05
Find the ratio of the large cube's volume to the small cube's volume
Now we can find the ratio by dividing the large cube's volume by the small cube's volume:
Ratio = V_L / V_s = (8 * x^3) / (x^3). The x^3 terms cancel out, leaving us with the answer: Ratio = 8.
06
Choose the correct answer
The volume of the large cube is 8 times the volume of the small cube. So, the correct answer is:
D. 8
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Volume Calculations
Understanding volume calculations is essential for solving many mathematical and real-world problems. The volume of an object is a measure of how much space that object occupies. For 3-dimensional shapes like cubes, calculating volume is straightforward once you know the formulas.
For a cube, which has all sides of equal length, the volume is found by raising the length of one edge to the third power, expressed mathematically as \( V = s^3 \), where \( V \) is the volume and \( s \) is the length of a side. This formula is derived from the basic principle that volume is the product of the area of the base times the height, and for a cube, the base area and height are the same.
When comparing two cubes, as seen in the exercise, the change in volume can be easily determined by scaling the edge length. If the edge of one cube is \( x \) and the edge of the larger cube is \( 2x \) (twice as long), using the volume formula reveals that the volume scales by the cube of the factor of increase (in this case, \( 2^3 \) or 8 times larger). This cube of the scaling factor for the edge lengths is a fundamental concept in volume scaling.
For a cube, which has all sides of equal length, the volume is found by raising the length of one edge to the third power, expressed mathematically as \( V = s^3 \), where \( V \) is the volume and \( s \) is the length of a side. This formula is derived from the basic principle that volume is the product of the area of the base times the height, and for a cube, the base area and height are the same.
When comparing two cubes, as seen in the exercise, the change in volume can be easily determined by scaling the edge length. If the edge of one cube is \( x \) and the edge of the larger cube is \( 2x \) (twice as long), using the volume formula reveals that the volume scales by the cube of the factor of increase (in this case, \( 2^3 \) or 8 times larger). This cube of the scaling factor for the edge lengths is a fundamental concept in volume scaling.
Geometric Properties
Geometric properties underpin the conceptual understanding necessary to solve problems involving shapes and volumes. In the context of cubes, the intrinsic properties include all sides being equal in length and all angles being right angles. A cube is a special case of a prism and belongs to the family of polyhedra.
Key geometric property for a cube is its symmetry. Cubes have a high degree of spatial symmetry, making it easier to understand scaling effects. When one dimension is scaled, due to the homogeneous shape of a cube, all three dimensions (length, width, and height) are scaled equally. Moreover, due to this property, when dealing with cube volume ratios, it is only necessary to understand how a change in one dimension affects the overall volume, since the same change applies to all dimensions.
When cubes are involved in mathematics problems, it's important to visualize that doubling one edge length increases the overall size significantly—not just doubling the volume as one might mistakenly assume without understanding geometric properties and the three-dimensional nature of volume.
Key geometric property for a cube is its symmetry. Cubes have a high degree of spatial symmetry, making it easier to understand scaling effects. When one dimension is scaled, due to the homogeneous shape of a cube, all three dimensions (length, width, and height) are scaled equally. Moreover, due to this property, when dealing with cube volume ratios, it is only necessary to understand how a change in one dimension affects the overall volume, since the same change applies to all dimensions.
When cubes are involved in mathematics problems, it's important to visualize that doubling one edge length increases the overall size significantly—not just doubling the volume as one might mistakenly assume without understanding geometric properties and the three-dimensional nature of volume.
Math Problem Solving
Math problem solving is not just about applying formulas—it involves understanding relationships and patterns. The approach to solving ratio problems, like finding how many times greater the volume of one cube is to another when their edges are scaled, starts with defining the relationship, as seen in the given steps.
The process of problem-solving often requires breaking down a problem into simpler parts. First, identify the given information and what is being asked. Next, find appropriate formulas and apply them accordingly. In the context of this problem, scaling the size of a cube's edge evenly scales its volume by the cube of the scale factor, providing a clear path to the solution. Problem-solving also involves recognizing that some factors cancel out, simplifying the calculation, as seen when the \( x^3 \) terms are eliminated in the ratio calculation. These steps emphasize logical thinking, which applies across various mathematical problems.
In summary, good math problem-solving integrates understanding from different areas of mathematics. For instance, it connects the geometric property of a cube to volume calculations—demonstrating a pattern that scaling a dimension of a cube will scale its volume exponentially.
The process of problem-solving often requires breaking down a problem into simpler parts. First, identify the given information and what is being asked. Next, find appropriate formulas and apply them accordingly. In the context of this problem, scaling the size of a cube's edge evenly scales its volume by the cube of the scale factor, providing a clear path to the solution. Problem-solving also involves recognizing that some factors cancel out, simplifying the calculation, as seen when the \( x^3 \) terms are eliminated in the ratio calculation. These steps emphasize logical thinking, which applies across various mathematical problems.
In summary, good math problem-solving integrates understanding from different areas of mathematics. For instance, it connects the geometric property of a cube to volume calculations—demonstrating a pattern that scaling a dimension of a cube will scale its volume exponentially.
