Calculating Volume of a Solid by Subtracting Volumes of Basic Shapes

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SUMMARY

The discussion focuses on calculating the volume of a solid formed by subtracting the volume of a hemisphere from that of a cylinder. The volume of the cylinder is calculated using the formula \( V_C = \pi a^2 h \), where \( a \) is the radius and \( h \) is the height. The volume of the hemisphere is given by \( V_H = \frac{2}{3} \pi a^3 \). The final volume of the solid is determined by the equation \( V_S = V_C - V_H \), which simplifies to \( V_S = \pi a^2 h - \frac{2}{3} \pi a^3 \).

PREREQUISITES
  • Understanding of geometric volume calculations
  • Familiarity with the formulas for cylinder and hemisphere volumes
  • Basic algebra for manipulating equations
  • Knowledge of mathematical constants such as π (pi)
NEXT STEPS
  • Study the derivation of volume formulas for various geometric shapes
  • Learn about integration techniques for calculating volumes of irregular solids
  • Explore applications of volume calculations in real-world scenarios
  • Investigate software tools for 3D modeling and volume calculations
USEFUL FOR

Students in geometry or calculus courses, educators teaching volume calculations, and professionals in fields requiring geometric analysis, such as engineering and architecture.

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Please solve this problem

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:)
 

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mathlearn said:
Please solve this problem

:)

Good evening and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

It is also MHB policy to not answer questions which are for credit or for exams.

Can you post what you have done so far?
 
Thank you super Sonic

So far

I calculated the volume of the cyclinder by taking "a" as the radius π(pie)a^2h

and I calculated the volume of the hemisphere (4/3πr^3)

and that was so far
 
Okay, using your formulas, we have the volume of a cylinder having radius $a$ as:

$$V_C=\pi a^2h$$

And the volume of a hemisphere of radius $a$ as:

$$V_H=\frac{2}{3}\pi a^3$$

Now, in order to find the volume of the given solid, we need to take the volume of the cylinder and remove (subtract) the volume of the hemisphere:

$$V_S=V_C-V_H$$

So, plug in for $V_C$ and $V_H$, then factor...what do you get?
 

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