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? 20 cm 30 cm 10 cm 15 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 \( V_{cone} = \frac{1}{3}\pi r^2 h_{cone} \). Here, the cylinder and cone have the same base radius (\( r \)) and same height (\( h = 30 \, \text{cm} \)).

Step2: Relate Volumes

Let the height of water in the cylinder be \( h_w \). The volume of water (from the cone) is equal to the volume of water in the cylinder. So, \( V_{cone} = V_{cylinder - water} \). Substituting the formulas: \( \frac{1}{3}\pi r^2 \times 30=\pi r^2 \times h_w \).

Step3: Solve for \( h_w \)

Cancel \( \pi r^2 \) from both sides: \( \frac{1}{3} \times 30 = h_w \). Calculate \( \frac{30}{3}=10 \, \text{cm} \).

Answer:

10 cm