Answer:
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
<em>The volume of the sphere is equal to</em>
we have
-----> the radius is half the diameter
substitute
<em>The surface area of the sphere is equal to</em>
substitute
therefore
The volume of the sphere is greater than the surface area of the sphere