# calculus

three bowls are filled to a depth of four units. a paraboloid, z=x^2 +y^2 for 0<z<4, a cone z=(x^2 + y^2)^(1/2) for 0<z<4, and a hyperboloid z=(1+x^2+y^2)^(1/2) for 1<z<5. which bowl holds more water?

1. 👍
2. 👎
3. 👁
1. Since the radius of the surface of the water is r^2 = x^2+y^2, it looks to me like the volumes are

paraboloid: v = ∫[0,4]πz dz = 8π
cone: v = ∫[0,4]πz^2 dz = 64/3 π
hyperboloid: v = ∫[1,5]π(z^2-1) dz = 112/3 π

Hmm. Not what I expected, the cone having greater volume than a round-bottomed paraboloid. However, it makes sense when you see that the cone's radius at the top of the water is √8, while the paraboloid's is only 2.

1. 👍
2. 👎

## Similar Questions

1. ### Algebra II

I'm confused.. 1. The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cable are 600 feet apart and 80 feet high. If the cables touch the road surface midway between the towers, what is the

2. ### Math

The base of a pyramid-shaped tank is a square with sides of length 12 feet, and the vertex of the pyramid is 10 feet above the base. The tank is filled to a depth of 4 feet, and water is flowing into the tank at the rate of 2

3. ### Chemistry

A snorkeler takes a syringe filled with 16 mL of air from the surface, where the pressure is 1.0 ATM, to an unknown depth. The volume of the air in the syringe at this depth is 7.5 mL. What Is the pressure at this depth? If the

4. ### AP calculus

The base of a cone-shaped tank is a circle of radius 5 feet, and the vertex of the cone is 12 feet above the base. The tank is being filled at a rate of 3 cubic feet per minute. Find the rate of change of the depth of water in the

1. ### math

Ali is making pottery. He uses 3/8 of a 2-pound bag of clay to make a set of bowls. Each set has 4 bowls. Drag one expression next to each question. The correct expression for each row is a way to answer that question. CLEAR CHECK

2. ### Maths

May someone help me with this HW problem? The depth (D metres) of water in a harbour at a time (t hours) after midnight on a particular day can be modelled by the function D = 2 sin(0.51t - 0.4) +5, t

3. ### algebra

At a pet store, Rhonda paid \$11.50 for 3 dog bowls and 4 bones. Kelly \$13 for 2 dog blows and 8 bones. How much will Andrew pay for 4 dog bowls and 2 bones?

4. ### Algebra 1 (Reiny or Kuai)

1. At a chili cook-off, people pay a \$3.00 entrance fee and \$0.50 for each bowl of chili they taste. The graph shows the total cost per person as a function of the number of bowls of chili tasted. a. Write a rule that gives the

1. ### Math

Ben sells 7/12 of his pottery. Of the remaining pottery, 3/5 are vases and the rest are bowls. What fraction of the remaining pottery are unsold bowls (bowls that remain)?

2. ### math

A packet of soup makes 9 bowls of serving . How many 3/4 bowls of serving are possible ?