Answer:
Part A) The volume of the entire cone is [tex]168\pi\ in^3[/tex]
Part B) see the explanation
Step-by-step explanation:
Part A) we know that
The volume of a cone is equal to
[tex]V=\frac{1}{3}\pi r^{2}h[/tex]
where
r is the radius of the base of the cone
h is the height of the cone
In this problem triangle ABD is similar to triangle ACE
Remember that If two figures are similar, then the ratio of its corresponding sides is proportional
so
[tex]\frac{AB}{AC}=\frac{BD}{CE}[/tex]
substitute the given values
[tex]\frac{7}{14}=\frac{3}{x}[/tex]
solve for x
[tex]x=14(3)/7\\x=6\ in[/tex]
To find out the volume of the entire cone we have
[tex]r=CE=x=6\ in[/tex]
[tex]h=AC=14\ in[/tex]
substitute in the formula
[tex]V=\frac{1}{3}\pi (6)^{2}(14)[/tex]
[tex]V=168\pi\ in^3[/tex]
Part B) How did you determine the value for x in triangle ACE
In this problem triangle ABD is similar to triangle ACE
If two figures are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent
so
[tex]\frac{AB}{AC}=\frac{BD}{CE}[/tex]
substitute the given values and solve for x
