Doubling the radius of a cone makes its volume 4 times greater.
D
d
V_(doubled)=4V
Practice makes perfect
a
We can calculate the volume of a cone with the radius r and the height h using the formula below.
V=1/3Ï€ r^2 hFrom the diagram we know that the height of the cone is 3. Substituting it into the formula, we can write as follows.
V=1/3π r^2( 3) ⇒ V=π r^2
We are asked to consider the radius values between 1 and 8. Let's start with the radius of 1. Then, doubling it we get 2. Again doubling this radius we get 4. After the last doubling, we get the radius of 8. We will calculate the volume of cones with these numbers as the radius. Let's make a table.
Radius
V=Ï€ r^2
Volume
r= 1
V=Ï€ ( 1)^2
V=Ï€
r= 2
V=Ï€ ( 2)^2
V=4Ï€
r= 4
V=Ï€ ( 4)^2
V=16Ï€
r= 8
V=Ï€ ( 8)^2
V=64Ï€
b
Using the table from Part A, we can form the following ordered pairs.
(1,Ï€), (2,4Ï€), (4,16Ï€), (8,64Ï€)
Let's plot them on the graph, where the x-axis represents the values of the radius and the y-axis represents the volume.
c
Let's analyze the volumes of the cones with the radii of 1 and 2.
r=1 ⇒ V=π
r=2 ⇒ V=4π
As we can see, the volume of the cone with the doubled radius is 4 times greater. Therefore, we can assume that doubling the radius of a cone makes its volume 4 times greater.
d
The volume of the cone with the radius r and the height h can be calculated using the formula below.
V=1/3Ï€ r^2 h
Let's use it to calculate the volume of the cone with the doubled radius.
r_(new)=2r
We can substitute it into the formula and solve for V_(new).
Let's compare the volumes of the cones with the radius of r and 2r.
l V= 1/3π r^2 h V_(new)=4* 1/3π r^2 h } ⇒ V_(new)=4V
The volume of the cone with the doubled radius is 4 times greater.
