Sure, let's solve this step-by-step.
### Step 1: Understanding the Volume of the Sphere
The volume of a sphere is given by the formula:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
However, we've already been given the volume of the sphere, which is 45π cm³.
### Step 2: The Volume of the Cylinder
When the metallic sphere is melted to form a cylinder of height 5 cm, the material (and therefore the volume) is conserved. Hence, the volume of the cylinder will be equal to the volume of the sphere.
The volume [tex]\( V_{\text{cylinder}} \)[/tex] of a cylinder is given by:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
Given:
[tex]\[ V_{\text{cylinder}} = 45\pi \, \text{cm}^3 \][/tex]
[tex]\[ h = 5 \, \text{cm} \][/tex]
### Step 3: Equating the Volumes
Setting the volume of the cylinder equal to the volume of the sphere, we get:
[tex]\[ \pi r^2 h = 45\pi \][/tex]
### Step 4: Simplifying for the Radius
We can divide both sides of the equation by π:
[tex]\[ r^2 h = 45\][/tex]
Now, we know the height [tex]\( h \)[/tex] of the cylinder is 5 cm, so:
[tex]\[ r^2 \cdot 5 = 45 \][/tex]
Divide both sides by 5:
[tex]\[ r^2 = \frac{45}{5} \][/tex]
[tex]\[ r^2 = 9 \][/tex]
### Step 5: Solving for the Radius
To find [tex]\( r \)[/tex], we take the square root of both sides:
[tex]\[ r = \sqrt{9} \][/tex]
[tex]\[ r = 3 \, \text{cm} \][/tex]
### Summary
Thus, the radius of the cylinder formed by melting a metallic sphere of volume [tex]\( 45\pi \)[/tex] cm³ and having a height of 5 cm is:
[tex]\[ \boxed{3 \, \text{cm}} \][/tex]
