Let's analyze the volume changes for each shape based on their dimensions increasing.
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Cone's radius increases x 3: The volume \( V \) of a cone is given by \( V = \frac{1}{3} \pi r^2 h \). If the radius \( r \) increases by a factor of 3 (i.e., becomes \( 3r \)), the new volume \( V' \) becomes: \[ V' = \frac{1}{3} \pi (3r)^2 h = \frac{1}{3} \pi (9r^2) h = 9 \left( \frac{1}{3} \pi r^2 h \right) = 9V \] Therefore, the cone's volume will increase by a factor of 9.
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Cone's height increases x 3: If the height \( h \) increases by a factor of 3 (i.e., becomes \( 3h \)), the new volume \( V'' \) will be: \[ V'' = \frac{1}{3} \pi r^2 (3h) = 3 \left( \frac{1}{3} \pi r^2 h \right) = 3V \] Therefore, the cone's volume will increase by a factor of 3.
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Sphere's radius increases x 3: The volume \( V \) of a sphere is given by \( V = \frac{4}{3} \pi r^3 \). If the radius \( r \) increases by a factor of 3 (i.e., becomes \( 3r \)), the new volume \( V' \) becomes: \[ V' = \frac{4}{3} \pi (3r)^3 = \frac{4}{3} \pi (27r^3) = 27V \] Therefore, the sphere's volume will increase by a factor of 27.
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Cylinder's radius increases x 6: The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \). If the radius \( r \) increases by a factor of 6 (i.e., becomes \( 6r \)), the new volume \( V' \) becomes: \[ V' = \pi (6r)^2 h = \pi (36r^2) h = 36V \] Therefore, the cylinder's volume will increase by a factor of 36.
Now, let's match the statements:
- A cone's radius increases x 3 → the cone's volume will increase x 9
- A cone's height increases x 3 → the cone's volume will increase x 3
- A sphere's radius increases x 3 → the sphere's volume will increase x 27
- A cylinder's radius increases x 6 → the cylinder's volume will increase x 36
So, the complete matching will be:
- a cone's radius increases x 3 → the cone's volume will increase x 9
- a cone's height increases x 3 → the cone's volume will increase x 3
- a sphere's radius increases x 3 → the sphere's volume will increase x 27
- a cylinder's radius increases x 6 → the cylinder's volume will increase x 36