volume of a cone = [tex]$\frac{1}{3} \pi r^2 h$[/tex], where [tex]$r$[/tex] is the radius and [tex]$h$[/tex] is the height.
The radius of the cone below is 6 m and its height is 11 m.
Work out the volume of the cone. Give your answer in terms of [tex]$\pi$[/tex].
Answer (1)
Substitute the given radius r = 6 m and height h = 11 m into the volume formula: V = 3 1 π r 2 h .
Calculate r 2 = 6 2 = 36 .
Multiply the values: V = 3 1 π ( 36 ) ( 11 ) = 3 1 π ( 396 ) .
Divide by 3 to find the volume: V = 132 π .
The volume of the cone is 132 π .
Explanation
Problem Analysis We are given the formula for the volume of a cone: V = 3 1 π r 2 h , where r is the radius and h is the height. We are also given that the radius of the cone is r = 6 m and the height is h = 11 m. Our goal is to find the volume of the cone in terms of π .
Substitute Values Now, we substitute the given values of r and h into the formula for the volume of a cone:
V = 3 1 π ( 6 2 ) ( 11 )
Calculate 6 squared First, we calculate 6 2 :
6 2 = 36
So, the equation becomes:
V = 3 1 π ( 36 ) ( 11 )
Multiply 36 by 11 Next, we multiply 36 by 11:
36 × 11 = 396
So, the equation becomes:
V = 3 1 π ( 396 )
Divide 396 by 3 Finally, we divide 396 by 3:
3 396 = 132
So, the volume of the cone is:
V = 132 π cubic meters.
Final Answer Therefore, the volume of the cone is 132 π cubic meters.
Examples
Cones are common shapes in everyday life, from ice cream cones to traffic cones. Knowing how to calculate the volume of a cone can be useful in various practical situations. For example, if you're filling ice cream cones at a shop, you might want to know the volume of each cone to ensure you're using the right amount of ice cream. Similarly, in construction or engineering, calculating the volume of conical piles of sand or gravel is essential for estimating material quantities. The formula for the volume of a cone, V = 3 1 π r 2 h , allows us to determine the space occupied by a cone, given its radius r and height h .
