A right rectangular prism has edges of 9 inches, 10 inches, and 7 inches. How many cubes with side lengths of 0.5 inches would be needed to fill the prism?
Understand the Problem
This problem involves finding the volume of a right rectangular prism (also called a right cuboid) and the volume of a smaller cube, and then determining how many of the smaller cubes fit into the prism. Specifically, we are given the dimensions of the prism as 9 inches by 10 inches by 7 inches, giving a volume of 9 * 10 * 7 = 630 cubic inches. Then we are told the side length of a cube is 0.5 inches, giving a volume of 0.5 * 0.5 * 0.5 = 0.125 cubic inches. To find the number of small cubes needed to fill the prism, we divide the volume of the prism by the volume of the small cube.
Answer
5040
Answer for screen readers
5040 cubes
Steps to Solve
- Calculate the volume of the right rectangular prism
To find the volume of a right rectangular prism, multiply its length, width, and height. $$V_{prism} = l \times w \times h = 9 \times 10 \times 7$$
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Simplify the volume of the prism $$V_{prism} = 630 \text{ cubic inches}$$
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Calculate the volume of the cube
To find the volume of a cube, raise the side length to the power of 3. $$V_{cube} = s^3 = (0.5)^3$$
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Simplify the volume of the cube $$V_{cube} = 0.125 \text{ cubic inches}$$
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Divide the volume of the prism by the volume of the cube to find how many cubes fit into the prism $$\text{Number of cubes} = \frac{V_{prism}}{V_{cube}} = \frac{630}{0.125}$$
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Simplify the fraction $$\text{Number of cubes} = 5040$$
5040 cubes
More Information
A right rectangular prism is a 3D shape with six rectangular faces where each face is perpendicular to its adjacent faces. Cubes are special cases of right rectangular prisms where all sides are of equal length.
Tips
A common mistake is to divide the dimensions of the prism by the side length of the cube individually (i.e. $9/0.5$, $10/0.5$, and $7/0.5$) and then multiplying those results together. This method will get to the correct answer, but students should not use this method because it lacks generalizability to more complex shapes. Another common mistake is making an error in calculating the volume of either the prism or the cube due to incorrect multiplication. Double-checking these calculations can prevent this.
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