A right circular cone has a volume of 140 in^3. The height of the cone is the same length as the diameter of the base. Find the radius and height.
We know the formula for the Volume of a Right Circular Cone is given by
V=(1/3)*pi*r^2*h
V=140 in^3
The height of the cone = diameter of the base. The diameter = 2 times the radius, so h = 2r
The formula for Volume can now be written as
V=(1/3)*pi*r^2*(2r)
which simplifies to
V=(2/3)*pi*r^3
You plug in 140 in^3 for V and solve for r. Then you can plug the value you find for r into the equation h=2r
V = (1/3 π r^2 h , but h = 2r
3V = π r^2 (2r) = 2π r^3
420 = 2πr^3
r^3 = 210/π
r = (210/π)^(1/3) = 4.0584
r = 4.0584
h = 8.11683
check:
V = (1/3)π(4.0584)^2 (8.11683) = 139.9989.. , not bad
To find the radius and height of the right circular cone, we can use the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
Given that the volume (V) is 140 in^3, and the height (h) is the same length as the diameter of the base, we can substitute these values into the formula:
140 = (1/3) * π * r^2 * h
Next, let's use the fact that the height (h) is equal to twice the radius (r):
h = 2r
We can substitute this into the volume formula:
140 = (1/3) * π * r^2 * 2r
Simplifying the equation, we get:
140 = (2/3) * π * r^3
To find the radius (r), we can isolate it on one side of the equation:
r^3 = (140 * 3) / (2 * π)
r^3 = 210 / π
Taking the cube root of both sides, we find:
r ≈ 3.378
Now that we have the radius, we can find the height (h) using the equation:
h = 2r
h ≈ 2 * 3.378
h ≈ 6.756
Therefore, the radius of the cone is approximately 3.378 inches and the height is approximately 6.756 inches.
To solve this problem, we need to use the formula for the volume of a right circular cone, which is given by:
V = (1/3) * π * r^2 * h
where V is the volume, r is the radius, and h is the height of the cone.
We are given the volume, V, as 140 in^3 and we know that the height, h, is the same length as the diameter of the base, which means h = 2r.
Substituting these values into the volume formula, we have:
140 = (1/3) * π * r^2 * (2r)
Simplifying the equation, we get:
140 = (2/3) * π * r^3
Next, we can solve for the radius, r:
r^3 = (140 * 3) / (2 * π)
r^3 = 210 / π
Taking the cube root of both sides:
r = (210 / π)^(1/3)
Now we can calculate the value of the radius, r.
Once we have the value of r, we can find the height, h, by substituting the value of r into the equation h = 2r.
Let's calculate the radius and the height: