Question

a chemist has an empty cylinder with a height of 30 cm and a cone - shaped flask. the flask has the same height and a base that is the same size as the cylinder’s. the flask is filled with water, which she pours into the cylinder. to what height does the water fill the cylinder? 15 cm 20 cm 10 cm 30 cm

Explanation:

Step1: Recall volume formulas

The volume of a cylinder is \( V_{cylinder} = \pi r^2 h_{cylinder} \), and the volume of a cone is \( V_{cone} = \frac{1}{3}\pi r^2 h_{cone} \). Given that the cylinder and the cone have the same base (so same radius \( r \)) and same height (\( h_{cylinder}=h_{cone} = 30\space cm \)).

Step2: Relate volumes

Let the radius of the base be \( r \) and the height of both be \( h = 30\space cm \). The volume of the cone (flask) is \( V_{cone}=\frac{1}{3}\pi r^2\times30 = 10\pi r^2 \). When we pour this volume into the cylinder, let the height of water in the cylinder be \( h_{water} \). The volume of water in the cylinder is \( V_{water}=\pi r^2 h_{water} \). Since the volume of water is the same as the volume of the cone, we set \( \pi r^2 h_{water}=\frac{1}{3}\pi r^2\times30 \).

Step3: Solve for \( h_{water} \)

We can cancel out \( \pi r^2 \) from both sides of the equation \( \pi r^2 h_{water}=\frac{1}{3}\pi r^2\times30 \), which gives \( h_{water}=\frac{1}{3}\times30 = 10\space cm \).

Answer:

10 cm