One right circular cone is set inside a larger circular cone. The cones share the same Axis, the same vertex, and the same height. Find the volume of the space between the cones of the diameter of the inside cone is 6in., the diameter of the outside cone is 9in and the height of both is 5 in. Round to the nearest tenth.
The surface area of a cone is 16.8pi in squared. The radius 3 in. What is the slant height?
for the volumes, just subtract the smaller from the larger:
π/3 (81)(5) = π/3 (36)(5) = 75π
area = πr(r+s)
16.8π = π(3)(3+s)
5.6=3+s
s=2.6
s cannot be less than r.
Evidently you do not want to include the base of the cone, so
16.8π = πrs
16.8π = 3πs
s = 5.6
To find the volume of the space between the cones, we can subtract the volume of the inner cone from the volume of the outer cone.
Let's calculate the volume of the outer cone first:
The formula for the volume of a cone is given by: V = (1/3) * π * r^2 * h,
where V is the volume, π is a constant approximately equal to 3.14159, r is the radius of the base of the cone, and h is the height of the cone.
For the outer cone:
Radius (r) = (9 in) / 2 = 4.5 in
Height (h) = 5 in
Plugging these values into the volume formula, we get:
V_outer = (1/3) * π * (4.5 in)^2 * 5 in
≈ 106.103 in^3 (rounded to the nearest thousandth)
Now, let's calculate the volume of the inner cone:
For the inner cone:
Radius (r) = (6 in) / 2 = 3 in
Height (h) = 5 in
Plugging these values into the volume formula:
V_inner = (1/3) * π * (3 in)^2 * 5 in
≈ 47.123 in^3 (rounded to the nearest thousandth)
Finally, we can find the volume of the space between the cones by subtracting the volume of the inner cone from the volume of the outer cone:
V_space = V_outer - V_inner
≈ 106.103 in^3 - 47.123 in^3
≈ 58.98 in^3 (rounded to the nearest hundredth)
Therefore, the volume of the space between the cones is approximately 58.98 cubic inches.
To find the volume of the space between the cones, we need to calculate the volume of the outer cone and the volume of the inner cone, and then subtract the volume of the inner cone from the volume of the outer cone.
Let's start by calculating the volume of the outer cone.
The formula for the volume of a cone is:
V = (1/3) * π * r² * h,
where V is the volume, π is a constant approximately equal to 3.14159, r is the radius, and h is the height.
Given that the diameter of the outside cone is 9 inches, the radius of the outside cone (R) is half of the diameter, so R = 9/2 = 4.5 inches.
Plugging in the values into the formula:
V_outer = (1/3) * π * (4.5)^2 * 5
Next, let's calculate the volume of the inner cone.
Given that the diameter of the inside cone is 6 inches, the radius of the inside cone (r) is half of the diameter, so r = 6/2 = 3 inches.
Plugging in the values into the formula:
V_inner = (1/3) * π * (3)^2 * 5
Finally, let's subtract the volume of the inner cone from the volume of the outer cone to get the volume of the space between the cones:
V_space = V_outer - V_inner
Calculating each part of the equation:
V_outer = (1/3) * 3.14159 * (4.5)^2 * 5
V_inner = (1/3) * 3.14159 * (3)^2 * 5
V_space = V_outer - V_inner
After performing the calculations, the volume of the space between the cones is approximately 70.7 cubic inches (rounded to the nearest tenth).