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? 30 cm 15 cm 10 cm 20 cm
The Correct Answer and Explanation is:
Correct Answer: 10 cm
Explanation:
We are given:
- A cylinder and a cone-shaped flask that have:
- The same height: 30 cm
- The same base area
- The cone-shaped flask is filled with water and emptied into the cylinder.
- We need to find to what height the water will fill the cylinder.
Step-by-step Reasoning:
Step 1: Use volume formulas
The volume of a cylinder is given by: Vcylinder=πr2hV_{\text{cylinder}} = \pi r^2 h
The volume of a cone is: Vcone=13πr2hV_{\text{cone}} = \frac{1}{3} \pi r^2 h
Because the cone and the cylinder have the same radius and same height, we can directly compare volumes.
Let’s suppose the radius is rr and the height is 3030 cm.
Then:
- Volume of the cylinder: πr2×30\pi r^2 \times 30
- Volume of the cone: 13πr2×30=πr2×10\frac{1}{3} \pi r^2 \times 30 = \pi r^2 \times 10
So, the volume of the cone is 1/3 the volume of the cylinder.
Step 2: Volume of water in the cylinder
The entire volume of the cone-shaped flask (water) is poured into the empty cylinder.
So, water fills: 13 of the cylinder’s height\frac{1}{3} \text{ of the cylinder’s height} Height filled=13×30 cm=10 cm\text{Height filled} = \frac{1}{3} \times 30 \text{ cm} = 10 \text{ cm}
Final Answer:
10 cm
This means the cylinder will be filled to a height of 10 cm with the water from the cone-shaped flask.
