A cylinder of radius 3 cm is carved out of a cylinder with a radius of 7 cm. If the two cylinders have the same height, find the volume of the remaining solid. What are the dimensi... A cylinder of radius 3 cm is carved out of a cylinder with a radius of 7 cm. If the two cylinders have the same height, find the volume of the remaining solid. What are the dimensions of each cylinder? How are their volumes related to each other?
Understand the Problem
The question asks to find the volume of the remaining solid after a smaller cylinder is carved out of a larger one. It provides specific dimensions and hints to help solve it.
Answer
The volume of the remaining solid is $V_{\text{remaining}} = 40\pi h$ cm³, where $h$ is the height of the cylinders.
Answer for screen readers
The volume of the remaining solid is given by $V_{\text{remaining}} = 40\pi h$ cm³, where $h$ is the height of the cylinders.
Steps to Solve
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Identify the dimensions of the cylinders
The radius of the larger cylinder is $R = 7 , \text{cm}$ and the radius of the smaller cylinder is $r = 3 , \text{cm}$. Both cylinders have the same height, which we will denote as $h$. -
Write the formula for the volume of a cylinder
The volume $V$ of a cylinder is given by the formula:
$$ V = \pi r^2 h $$
We will use this formula to find the volumes of both cylinders. -
Calculate the volume of the larger cylinder
Using the larger cylinder's radius and height, the volume $V_{\text{large}}$ is:
$$ V_{\text{large}} = \pi (7)^2 h = 49\pi h $$ -
Calculate the volume of the smaller cylinder
Using the smaller cylinder's radius and height, the volume $V_{\text{small}}$ is:
$$ V_{\text{small}} = \pi (3)^2 h = 9\pi h $$ -
Find the volume of the remaining solid
To find the volume of the remaining solid, we subtract the volume of the smaller cylinder from the volume of the larger cylinder:
$$ V_{\text{remaining}} = V_{\text{large}} - V_{\text{small}} $$
Substituting the volume formulas gives:
$$ V_{\text{remaining}} = 49\pi h - 9\pi h = 40\pi h $$
The volume of the remaining solid is given by $V_{\text{remaining}} = 40\pi h$ cm³, where $h$ is the height of the cylinders.
More Information
This result shows the volume of the cylindrical solid that remains after carving out the smaller cylinder. The final volume will depend on the height $h$, which can be specified if known.
Tips
- Forgetting to use the correct formula for the volume of a cylinder. Always ensure to apply $V = \pi r^2 h$ correctly.
- Miscalculating the radius or height when substituting values, which can lead to incorrect volume calculations.
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