Calculating the volume of the cylinder:

To find the number of solid spheres that can be made from the given metal cylinder, we need to first calculate the volume of the cylinder.

The volume of a cylinder can be calculated using the formula:
V = π * r^2 * h

Where:
V = Volume of the cylinder
π = Pi (approximately 3.14159)
r = Radius of the cylinder (which is half of the diameter)
h = Height of the cylinder

Given that the diameter of the cylinder is 4 cm, the radius would be half of that, which is 2 cm. And the height of the cylinder is given as 45 cm.

Substituting these values into the formula, we get:
V = 3.14159 * 2^2 * 45 = 3.14159 * 4 * 45 = 565.4866 cm^3

So, the volume of the metal cylinder is 565.4866 cubic centimeters.

Calculating the volume of a sphere:

To find the number of solid spheres that can be made from the cylinder, we need to calculate the volume of a single sphere.

The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3

Where:
V = Volume of the sphere
π = Pi (approximately 3.14159)
r = Radius of the sphere (which is half of the diameter)

Given that the diameter of the sphere is 6 cm, the radius would be half of that, which is 3 cm.

Substituting these values into the formula, we get:
V = (4/3) * 3.14159 * 3^3 = (4/3) * 3.14159 * 27 = 113.097 cm^3

So, the volume of a single sphere is 113.097 cubic centimeters.

Calculating the number of spheres:

To find the number of spheres that can be made from the cylinder, we divide the volume of the cylinder by the volume of a single sphere.

Number of spheres = Volume of cylinder / Volume of a single sphere

Substituting the values we calculated earlier, we get:
Number of spheres = 565.4866 / 113.097 = 5

So, the number of solid spheres, each with a diameter of 6 cm, that can be made from the given metal cylinder is 5.