Question Video: Finding the Volume of a Compound Solid Involving a Cylinder and Hemispheres | Nagwa Question Video: Finding the Volume of a Compound Solid Involving a Cylinder and Hemispheres | Nagwa

Question Video: Finding the Volume of a Compound Solid Involving a Cylinder and Hemispheres Mathematics

The shape in the figure consists of a cylinder with a hemisphere attached to each end. Work out its volume, giving your answer to two decimal places.

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Video Transcript

The shape in the given figure consists of a cylinder with a hemisphere attached to each end. Work out its volume, giving your answer to two decimal places.

We’re told in the question, but we can also see from the diagram, that this composite shape consists of a cylinder and two hemispheres. We can see that these two hemispheres are congruent because they each have a radius of three feet. So, the total volume will be equal to the volume of the cylinder plus twice the volume of the hemisphere. Two identical hemispheres though make a sphere. So, we can simplify slightly by calculating the volume of the cylinder and the volume of a sphere.

Let’s consider the cylinder first of all then. We know that its volume is calculated using the formula 𝜋𝑟 squared ℎ. That’s the cross-sectional area multiplied by the height of the cylinder. From the figure, we can see that the height of the cylinder is 10 feet, but what about its radius? Well, it’s just the same as the radius of the hemisphere on each end, so it’s three feet. The volume of the cylinder is, therefore, 𝜋 multiplied by three squared multiplied by 10. That simplifies to 90𝜋. And we’ll keep our answer in terms of 𝜋 for now.

For the two hemispheres, which we’ve already said we can model as a single sphere, the volume is given by four-thirds 𝜋𝑟 cubed. We, therefore, have four-thirds multiplied by 𝜋 multiplied by three cubed. Three cubed is equal to 27. And we can then cancel a factor of three from the numerator and denominator. We’re left with four multiplied by 𝜋 multiplied by nine, which is 36𝜋. The total volume of the shape in the figure then is 90𝜋 for the volume of the cylinder plus 36𝜋 for the volume of the sphere, or two hemispheres, which is 126𝜋.

This would be a perfectly acceptable format for our answer, and indeed, it’s an exact value. But the question asked for the answer to two decimal places. So, evaluating this on a calculator, and we have 395.84067. Rounding appropriately and we have our answer to the problem, the units of which will be cubic feet. The total volume of the shape in the given figure to two decimal places is 395.84 cubic feet.

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