A right cubic prism has edges of 3 1/4 inches. How many cubes with side lengths of 1/4 inches would be needed to fill the prism?
Understand the Problem
The question asks how many cubes with side lengths of 1/4 inches are needed to fill a right cubic prism with edges of 3 1/4 inches. This involves calculating the volumes of both the prism and the smaller cubes, then dividing the prism's volume by the cube's volume to find the number of cubes needed.
Answer
2197
Answer for screen readers
2197
Steps to Solve
- Convert mixed numbers to improper fractions
Convert the side length of the right cubic prism from a mixed number to an improper fraction: $$3\frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}$$
- Calculate the volume of the right cubic prism
Since the prism is cubic, all sides are equal. Volume is side length cubed: $$V_{prism} = \left(\frac{13}{4}\right)^3 = \frac{13^3}{4^3} = \frac{2197}{64} \text{ cubic inches}$$
- Calculate the volume of one small cube
The side length of a small cube is $\frac{1}{4}$ inches. The volume is: $$V_{cube} = \left(\frac{1}{4}\right)^3 = \frac{1^3}{4^3} = \frac{1}{64} \text{ cubic inches}$$
- Determine the number of cubes needed
Divide the volume of the prism by the volume of one small cube to find out how many cubes fit into the prism: $$\text{Number of cubes} = \frac{V_{prism}}{V_{cube}} = \frac{\frac{2197}{64}}{\frac{1}{64}} = \frac{2197}{64} \times \frac{64}{1} = 2197$$
2197
More Information
It takes 2197 small cubes with side lengths of 1/4 inches to fill a right cubic prism with edges of 3 1/4 inches.
Tips
A common mistake is forgetting to cube the side lengths when calculating the volumes. Another mistake is incorrectly converting mixed numbers to improper fractions, which will lead to an incorrect volume calculation. Additionally, students might make arithmetic errors when cubing the fractions or when dividing the volumes.
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