Show that the volume of the largest right circular cylinder that can be inscribed in a given right circular cone is 4/9 the volume of the cone
draw a side view
If the cone has radius R and height H, and the cylinder has radius r and height h, then using similar triangles,
R/H = (R-r)/h
The volume of the cylinder is
v = πr^2 h = πr^2 * (R-r) * H/R = kr^2(R-r) for k=πH/R
dv/dr = 2Rr - 3r^2 = r(2R-3r)
dv/dr = 0 when r = 2/3 R
when r = 2/3 R, h = (R-r) * H/R = R/3 * H/R = H/3
So πr^2 h = π(2/3 R)^2*(H/3) = 4/9 * 1/3 πR^2 H
To find the volume of the largest right circular cylinder inscribed in a given right circular cone, we can use the methods of geometry and calculus. Let's break down the problem step by step.
Step 1: Define Variables
Let's label the given right circular cone. Let the height of the cone be 'h' and the radius of the cone be 'r'. Additionally, let's label the radius of the inscribed cylinder as 'R' and its height as 'H'.
Step 2: Draw a Diagram
Visualize the given right circular cone and the inscribed right circular cylinder. Draw the height of the cone, the radius of the cone, the height of the cylinder inside the cone, and the radius of the cylinder.
Step 3: Relate the Variables
The radius of the inscribed cylinder will touch the base of the cone. Hence, the radius 'R' of the cylinder will be equal to the radius 'r' of the cone.
Step 4: Determine the Relationship Between the Heights
Since the base circle of the cone is larger than the cylinder's base circle, the height 'H' of the cylinder will be less than the height 'h' of the cone. To find the relationship between the two heights, we use similar triangles.
In the given cone, we can create a right triangle with the height 'h', the radius 'r', and a slant height 's'. In the inscribed cylinder, we have a similar right triangle with the height 'H', the radius 'R' (which is equal to 'r'), and the slant height 'S'.
Using the similar triangles, we can set up the proportion:
(h - H) / h = R / r
Simplifying the equation, we get:
(h - H) / h = r / r
(h - H) / h = 1
h - H = h
H = 0
Therefore, the height of the inscribed cylinder is zero, meaning it touches the apex of the cone.
Step 5: Calculate the Volumes
The volume of the given right circular cone is given by the formula:
Volume of Cone = (1/3) * π * r^2 * h
The volume of the inscribed right circular cylinder is given by the formula:
Volume of Cylinder = π * R^2 * H
Since H = 0 (as derived in Step 4), the volume of the cylinder becomes:
Volume of Cylinder = π * R^2 * 0
Volume of Cylinder = 0
Step 6: Compare the Volumes
To find the ratio of the volume of the cylinder to the volume of the cone, we divide the volume of the cylinder by the volume of the cone:
Volume of Cylinder / Volume of Cone = 0 / ((1/3) * π * r^2 * h)
Simplifying the ratio, we get:
Volume of Cylinder / Volume of Cone = 0
This shows that the volume of the largest right circular cylinder that can be inscribed in a given right circular cone is zero, meaning the cylinder has no volume.
In conclusion, the volume of the largest right circular cylinder that can be inscribed in a given right circular cone is zero, which is not 4/9 of the volume of the cone.