A solid metallic hemisphere of radius 8 cm is melted and recast into a right circular cone of base radius 6 cm. Determine the height of the cone.
\(\displaystyle h=\dfrac{256}{9}\,\text{cm}\approx 28.44\,\text{cm}\)
Step 1: Understand the problem.
A metallic hemisphere is melted and reshaped into a cone. Since the material is the same, the volume of hemisphere = volume of cone.
Step 2: Write the formula for the volume of a hemisphere.
The volume of a sphere is \(V = \dfrac{4}{3}\pi r^3\).
A hemisphere is half a sphere, so:
\(V_{hemisphere} = \dfrac{1}{2} \times \dfrac{4}{3}\pi r^3 = \dfrac{2}{3}\pi r^3\).
Step 3: Substitute the radius of the hemisphere.
Here, \(r = 8\,\text{cm}\).
So, \(V_{hemisphere} = \dfrac{2}{3}\pi (8)^3 = \dfrac{2}{3}\pi (512) = \dfrac{1024}{3}\pi\,\text{cm}^3\).
Step 4: Write the formula for the volume of a cone.
The volume of a cone is \(V_{cone} = \dfrac{1}{3}\pi R^2 h\),
where \(R\) is the base radius, \(h\) is the height.
Step 5: Substitute the base radius of the cone.
Here, \(R = 6\,\text{cm}\).
So, \(V_{cone} = \dfrac{1}{3}\pi (6)^2 h = \dfrac{1}{3}\pi (36) h = 12\pi h\,\text{cm}^3\).
Step 6: Equate the volumes.
\(V_{hemisphere} = V_{cone}\)
\(\dfrac{1024}{3}\pi = 12\pi h\)
Step 7: Simplify the equation.
Cancel \(\pi\) on both sides:
\(\dfrac{1024}{3} = 12h\)
Multiply both sides by 3:
\(1024 = 36h\)
Divide both sides by 36:
\(h = \dfrac{1024}{36} = \dfrac{256}{9}\,\text{cm}\).
Step 8: Write the final answer in decimal form.
\(h \approx 28.44\,\text{cm}\).
Therefore, the height of the cone is \(28.44\,\text{cm}\).