Suppose you have a cup in the shape of a right circular cone. Fill it with water to half the depth of the cone. What fraction of the volume of the cup is occupied by the water?
Answer: The water forms a cone with half the height and half the radius, so its volume is (12)3 = 18 of the cup.
Step-by-step solution
Idea: The cup points down. Water up to half the depth is a smaller cone of the same shape; at half the height the width is also half (similar triangles).
- Let the cup have radius r and depth (height) h. Its volume is 13πr2h.½ mark
- The water is a cone with height h2. By similar triangles its radius is also halved: r2.1 mark
- Water volume = 13π(r2)2(h2) = 18 × 13πr2h.1 mark
- Fraction = 18. The cup still has 78 of its space empty, though it looks half full.½ mark
The water occupies 1/8 of the volume of the cup.
Check: Cup r = 4, h = 6: volume 32π. Water r = 2, h = 3: volume 4π. 4π ÷ 32π = 18 ✓.
Answer to write in the exam
Water cone: height h2, radius r2 (similar triangles)
Vwater = 13π(r2)2(h2) = 18 × 13πr2h
∴ Fraction = 18
Common mistakes that cost marks
- Answering 12 because the depth is half.
- Halving the height but keeping the full radius, which gives 12.
- Halving both but squaring neither: 14.
How this can come in the exam
MCQ (1 mark)
A conical funnel (point down) is filled with water to one-third of its depth. The fraction of its volume filled is
- 13
- 19
- 127
- 16
Show answer
(C) 127
Scale factor 13 in every direction: (13)3 = 127.
Try one yourself
A conical glass (point down) holds 240 mL when full. How much water is in it when it is filled to half its depth?
Show answer
240 × 18 = 30 mL.
More questions like this
- But you may be able to find your own derivation of the formula by connecting it with the formula for volume of a cone; namely, by writing it as
V = 13 × (4πr2) × r,
i.e., Volume of sphere = 13 × surface area of sphere × radius of sphere.
Try to work out the connecting links on your own. - Take a rubber ball and drive a nail into it. Using the nail for support, wind a string tightly around the ball. When you have reached the ‘fullest’ part of the ball, use pins to keep the string in place, and continue to wind the string around the remaining part of the ball, till you have completely covered it; see the figure. Mark the starting and finishing points on the string, and unwind the string from the surface of the ball. Now, measure the diameter of the ball and get its radius. On a sheet of paper, draw four circles with a radius equal to the radius of the ball. Fill the circles with the string you had wound around the ball (as shown in the figure).
- A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain?
- A ball bearing has a radius of 0.7 cm. Find its surface area.
- Two solid spheres made of the same metal have weights 5920 g and 740 g. Determine the radius of the larger sphere, if the diameter of the smaller one is 5 cm.