Volume of a Cone
The volume of a right circular cone is one-third of the volume of a right circular cylinder of the same base and same height.
$$\therefore $$ If $$h$$ is the height of the cone and $$r$$ is the radius of the base, then
\[{\text{Volume }} = {\text{ }}\frac{1}{3}{\text{ }} \times {\text{ area of the base }} \times {\text{ altitude}}\]
$$\therefore $$ $$V = \frac{1}{3}\pi {r^2}h$$ (as the area of the base $$ = \pi {r^2}$$)
Rule: The volume of a cone equals the area of the base times one-third the altitude.
Example:
The circumference of the base of a $$9$$m high conical tent is $$44$$m. Find the volume of the air contained in it.
Solution:
Circumference of the base $$ = 2\pi r = 44$$m
$$\therefore $$ $$2 \times \frac{{22}}{7} \times r = 44$$
$$r = \frac{{44 \times 7}}{{44}} = 7$$m
$$\because $$ height of the conical tent $$ = 9$$m
$$\therefore $$ volume of air $$ = \frac{1}{3}\pi {r^2}h$$
$$ = \frac{1}{3} \times \frac{{22}}{7} \times 7 \times 9 = 462$$ cubic m
Example:
The vertical height of a conical tent is $$42$$ dm and the diameter of its base is $$5.4$$ m. Find the number of people it can accommodate if each person is to be allowed $$2916$$ cubic dm of space.
Solution:
Here height $$h = 42$$ dm
Diameter $$ = 5.4$$ m $$ = 54$$ dm
Radius $$r = 27$$ dm
Volume $$ = \frac{1}{3}\pi {r^2}h$$
$$ = \frac{1}{3} \times \frac{{22}}{7} \times 27 \times 27 \times 42$$
$$ = 32076$$ Cubic dm
Space allowed for $$1$$ person $$ = 2916$$ cubic dm
$$\therefore $$ the required number of people $$ = \frac{{32076}}{{2916}} = 11$$ people
er
July 23 @ 9:27 am
there is a mistake with one of the solution, and it should not be 7 but instead should be 49
Bob
May 7 @ 1:11 am
I disagree eith the formula. It’s sin(pix diameter)