9) Volume of the cone:
10) New volume of the cone:
11) The volume of the cone shrinks by a factor of
12) Volume of the composite figure:
Step-by-step explanation:
9)
The volume of a cone is given by the equation
where
is the area of the base, where
r is the radius of the cone
h is the height of the cone
For the cone in this problem, we have:
r = 18 mm (radius)
h = 27 mm (height)
Therefore, the volume of this cone is
10)
In this second exercise, the radius and the height of the cone are divided by 3. Therefore we have:
is the new radius of the cone
is the new height of the cone
Therefore, the new volume of the cone is
And substituting, we find
11)
We can understand what happened to the volume of the cone after dividing its dimensions by 3 by calculating the ratio of the original volume to the new volume:
This means that the new volume of the cone is 27 times smaller than the original volume.
We see that this "factor of shrinking" corresponds to the total factor of shrinking ofits dimensions, to the power of 3. In fact:
- Each the radius and the height of the cone has shrunk by a factor of 3
- However, the cone is a figure in 3 dimensions, so the total factor of shrinking of its volume is
12)
The composite figure consists of a squared pyramid + a parallelepiped.
The volume of the square pyramid is given by:
where
L = 5 ft is the length of the base
w = 1 ft is the width of the base
h = 6 ft is the height of the pyramid
Substituting,
The volume of the parallelepiped is given by
where
L = 5 ft is the length of the base
w = 1 ft is the width of the base
h = 1 ft is the height
Substituting,
So, the total volume of the composite figure is
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