How do you calculate the volume of a cylinder combined with a hemisphere if the diameter of the base is 4 cm and the height of the cylinder is 25 cm?

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Calculating the Volume of a Cylinder with a Hemisphere

To find the total volume of the figure, which consists of a cylindrical part and a hemispherical part, we need to calculate the volume of each component separately and then sum them up.

Step 1: Calculate the Volume of the Cylinder

The formula for the volume of a cylinder is:

V_cylinder = πr²h

Where:

  • r = radius of the base
  • h = height of the cylinder

Given that the diameter of the base is 4 cm, the radius (r) will be:

r = diameter / 2 = 4 cm / 2 = 2 cm

The height (h) of the cylinder is given as 25 cm. Plugging these values into the formula:

V_cylinder = π(2 cm)²(25 cm)
    = π(4 cm²)(25 cm)
    = 100π cm³

Step 2: Calculate the Volume of the Hemisphere

The volume for a complete sphere is given by:

V_sphere = (4/3)πr³

Since we have a hemisphere, we take half of that volume:

V_hemisphere = (1/2) * (4/3)πr³ = (2/3)πr³

Using the same radius of 2 cm:

V_hemisphere = (2/3)π(2 cm)³
    = (2/3)π(8 cm³)
    = (16/3)π cm³

Step 3: Total Volume of the Figure

The total volume (V_total) is the sum of the cylinder’s volume and the hemisphere’s volume:

V_total = V_cylinder + V_hemisphere
    = 100π cm³ + (16/3)π cm³

To combine these, we first convert 100π to have a common denominator:

100π cm³ = (300/3)π cm³

Now we can add the volumes:

V_total = (300/3)π cm³ + (16/3)π cm³
    = (316/3)π cm³

Thus, the total volume of the figure, which is a cylinder with a hemisphere on top, is:

V_total ≈ 331.61 cm³ (using π ≈ 3.14)

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