1. **Problem statement:** Determine the effect on volume when dimensions of cubes, cylinders, and rectangular prisms are changed as described. 2. **Formulas:** - Volume of cube: $$V = s^3$$ where $s$ is side length. - Volume of cylinder: $$V = \pi r^2 h$$ where $r$ is radius and $h$ is height. - Volume of rectangular prism: $$V = lwh$$ where $l$, $w$, $h$ are length, width, height. 3. **Part a:** Side length of cube is tripled. - New side length: $3s$ - New volume: $$V' = (3s)^3 = 27s^3$$ - Effect: Volume is multiplied by 27. 4. **Part b:** Side length of cube is halved. - New side length: $\frac{s}{2}$ - New volume: $$V' = \left(\frac{s}{2}\right)^3 = \frac{s^3}{8}$$ - Effect: Volume is multiplied by $\frac{1}{8}$. 5. **Part c:** Radius doubled, height halved for cylinder. - New radius: $2r$ - New height: $\frac{h}{2}$ - New volume: $$V' = \pi (2r)^2 \times \frac{h}{2} = \pi \times 4r^2 \times \frac{h}{2} = 2\pi r^2 h$$ - Effect: Volume is multiplied by 2. 6. **Part d:** Radius doubled, height divided by 4 for cylinder. - New radius: $2r$ - New height: $\frac{h}{4}$ - New volume: $$V' = \pi (2r)^2 \times \frac{h}{4} = \pi \times 4r^2 \times \frac{h}{4} = \pi r^2 h$$ - Effect: Volume remains the same (multiplied by 1). 7. **Part e:** Length doubled, width halved, height tripled for rectangular prism. - New length: $2l$ - New width: $\frac{w}{2}$ - New height: $3h$ - New volume: $$V' = 2l \times \frac{w}{2} \times 3h = \cancel{2}l \times \frac{w}{\cancel{2}} \times 3h = 3lwh$$ - Effect: Volume is multiplied by 3. **Final answers:** - a) Volume multiplied by 27 - b) Volume multiplied by $\frac{1}{8}$ - c) Volume multiplied by 2 - d) Volume multiplied by 1 (no change) - e) Volume multiplied by 3