To determine the height to which the water fills the cylinder, we first need to understand the volumes of the cone-shaped flask and the cylinder.
Both the cylinder and the cone have the same base radius \( r \) and the same height \( h = 30 \text{ cm} \).
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Volume of the cylinder (V_cylinder): \[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (30) \]
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Volume of the cone (V_cone): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (30) = 10 \pi r^2 \]
Since the cone is filled with water, the volume of water in the cone is \( 10 \pi r^2 \).
Now, when the chemist pours this water into the cylinder, the volume of water in the cylinder will still be \( 10 \pi r^2 \).
Next, we need to determine how high \( h_{\text{water}} \) the water will fill the cylinder when it contains \( 10 \pi r^2 \) of water.
The volume of water in the cylinder when filled to height \( h_{\text{water}} \) is given by: \[ V_{\text{water}} = \pi r^2 h_{\text{water}} \]
Setting the two volumes equal gives us: \[ \pi r^2 h_{\text{water}} = 10 \pi r^2 \]
Dividing both sides by \( \pi r^2 \) (assuming \( r \neq 0 \)): \[ h_{\text{water}} = 10 \text{ cm} \]
Thus, the height to which the water fills the cylinder is 10 cm.