Free Math Worksheets

- The same number of small cubes must fit along each of the three dimensions. - What number cubed equals \(64\)? - Use a cube root to find the number of small cubes along one edge.

1. The same number of small cubes must fit along the length, width, and height of the large cube. 2. The number of cubes along each edge is \(\sqrt[3]{64}=4\), because \(4\cdot4\cdot4=64\). 3. Therefore, each edge of the large cube is \(4\) times the edge length of a small cube.

The ratio is \(4{:}1\). Each edge of the large cube is \(4\) times as long as an edge of a small cube.

- Use the relationship between a cube's volume and its edge length. - Find two consecutive perfect cubes that enclose \(60\). - Test the midpoint of the two possible whole-number estimates.

1. The edge length is \(\sqrt[3]{60}\,\text{dm}\). 2. Since \(3^3=27\) and \(4^3=64\), \(27<60<64\). Therefore, the edge length is between \(3\,\text{dm}\) and \(4\,\text{dm}\). 3. The midpoint between \(3\) and \(4\) is \(3.5\). Since \(3.5^3=42.875<60\), \(\sqrt[3]{60}>3.5\). 4. Therefore, the edge length is closer to \(4\,\text{dm}\) than to \(3\,\text{dm}\).

a) The edge length is between \(3\,\text{dm}\) and \(4\,\text{dm}\). b) \(4\,\text{dm}\) is the better estimate because \(3.5^3=42.875<60\), so \(\sqrt[3]{60}>3.5\).

- Take the cube root of each volume. - Compare the two edge lengths. - You can also find the cube root of the volume ratio.

1. The smaller cube's edge length is \(\sqrt[3]{8}=2\,\text{dm}\). 2. The larger cube's edge length is \(\sqrt[3]{216}=6\,\text{dm}\). 3. The ratio is \(6\div2=3\), so the larger cube's edge is \(3\) times as long. 4. Equivalently, the volume ratio is \(216\div8=27=3^3\), so the edge-length ratio is \(\sqrt[3]{27}=3\).

The larger cube's edge length is \(3\) times the smaller cube's edge length, so the ratio is \(3{:}1\).

- Convert the volume to cubic centimeters. - Find the cube root of the volume. - A cube has six congruent square faces.

1. Convert the volume: \(0.008\,\text{m}^3=8000\,\text{cm}^3\). 2. The edge length is \(\sqrt[3]{8000}=20\,\text{cm}\). 3. The surface area is \(6s^2=6(20^2)=2400\,\text{cm}^2\).

a) The edge length is \(20\,\text{cm}\). b) The surface area is \(2400\,\text{cm}^2\).

- Convert all measurements to centimeter-based units. - Use \(V=Bh\) for part a. - Compare the volume with nearby perfect cubes for part b.

1. Convert the measurements: \(1.44\,\text{dm}^3=1440\,\text{cm}^3\) and \(80\,\text{mm}=8\,\text{cm}\). 2. The base area is \(12\cdot8=96\,\text{cm}^2\), so the height is \(1440\div96=15\,\text{cm}\). 3. A cube with a whole-number edge has a perfect-cube volume. Since \(11^3=1331\) and \(12^3=1728\), and \(1331<1440<1728\), \(1440\) is not a perfect cube.

a) The prism's height is \(15\,\text{cm}\). b) No. The cube's edge cannot be a whole number of centimeters because \(1440\) lies between the consecutive perfect cubes \(11^3\) and \(12^3\).

- Convert both measurements to centimeter-based units. - Use a cube root to find Cube A's edge. - Use \(SA=6s^2\) to find Cube B's edge.

1. Convert Cube A's volume: \(0.008\,\text{dm}^3=8\,\text{cm}^3\). 2. Its edge length is \(\sqrt[3]{8}=2\,\text{cm}\). 3. For Cube B, \(6s^2=54\), so \(s^2=9\) and \(s=3\,\text{cm}\). 4. Cube B has the longer edge. Its volume is \(3^3=27\,\text{cm}^3\). 5. The volume difference is \(27-8=19\,\text{cm}^3\).

Cube B has the longer edge: \(3\,\text{cm}\), compared with \(2\,\text{cm}\) for Cube A. The volume difference is \(19\,\text{cm}^3\).