The ratio of the volumes of the similar solids is 25: 1 5: 1 125: 1
the ratio of the surface areas of the similar solids is 125: 1 5: 1 25: 1
the ratio of the heights of the similar 125: 1 5: 1 25: 1
two spheres with different radii measurements similar (always, never sometimes)
the length of the diameter of a sphere is 8 inches. the volume of the sphere the surface area of the sphere. (less than, greater than, equal to)

Part 1)

a) The ratio of the heights of the similar solids is 5/1

b) The ratio of the surface areas of the similar solids is (5/1)²=25/1

c)  The ratio of the volumes of the similar solids is (5/1)³=125/1

Part 2) Two spheres with different radii measurements are always similar

Part 3) The volume of the sphere is greater than the surface area of the sphere

Step-by-step explanation:

Part 1) we know that

The ratio of the corresponding heights of the similar solids is equal to the scale factor

The ratio of the surface areas of the similar solids is equal to the scale factor squared

The ratio of the volumes of the similar solids is equal to the scale factor elevated to the cube

In this problem

The scale factor is 5/1

therefore

a) The ratio of the heights of the similar solids is 5/1

b) The ratio of the surface areas of the similar solids is (5/1)²=25/1

c)  The ratio of the volumes of the similar solids is (5/1)³=125/1

Part 2)

we know that

Figures can be proven similar if one, or more, similarity transformations (reflections, translations, rotations, dilations) can be found that map one figure onto another.  

In this problem to prove that two spheres  are similar, a translation and a scale factor (from a dilation) will be found to map one sphere onto another.

therefore

Two spheres with different radii measurements are always similar

Part 3) The length of the diameter of a sphere is 8 inches

The volume of the sphere is equal to

we have

-----> the radius is half the diameter

substitute

The surface area of the sphere is equal to

substitute

therefore

The volume of the sphere is greater than the surface area of the sphere

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