Question
What is the formula for the volume of a cylinder with radius r and height h? In 3–5 sentences, describe an informal argument for this volume formula.(4 points)
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GPT-4o mini
The formula for the volume \( V \) of a cylinder with radius \( r \) and height \( h \) is given by:
\[ V = \pi r^2 h. \]
To understand why this formula works, consider that the volume of a cylinder can be thought of as stacking many thin circular disks (each with a very small thickness) on top of one another. The area of the base of the cylinder, which is a circle, is \( \pi r^2 \). Since the cylinder has height \( h \), we can visualize the total volume as the area of the base multiplied by the height—effectively adding up the areas of all those thin disks from the bottom to the top of the cylinder. This leads us straight to the formula \( V = \pi r^2 h \), where \( \pi r^2 \) represents the area of the base, and \( h \) represents the cumulative height of those disks.
\[ V = \pi r^2 h. \]
To understand why this formula works, consider that the volume of a cylinder can be thought of as stacking many thin circular disks (each with a very small thickness) on top of one another. The area of the base of the cylinder, which is a circle, is \( \pi r^2 \). Since the cylinder has height \( h \), we can visualize the total volume as the area of the base multiplied by the height—effectively adding up the areas of all those thin disks from the bottom to the top of the cylinder. This leads us straight to the formula \( V = \pi r^2 h \), where \( \pi r^2 \) represents the area of the base, and \( h \) represents the cumulative height of those disks.
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