In the figure, the radius of the sector is 12cm and ∠AOB = 150°. The sector is bent to form a conical vessel of base radius r cm. Water is poured into the vessel to a depth equal to half the height of the vessel. Find the volume of the water inside the vessel.
Figure http://aycu34.webshots.com/image/40073/2004771548809077161_rs.jpg
math problem~help
2007-12-30 3:03 am
回答 (2)
2007-12-30 3:54 am
✔ 最佳答案
(1) Set up this equation to solve for r
arc length of the sector = circumference of the base of the cone
(2)
Consider triangle OAC.
You know r and OB=12.
Use Pythagoras' Theorem to find OC.
(3)
Find volume of the cone.
(You know r and OC already)
(4)
The cone and water are similar figures.
So they have this relationship.
(height of cone / height of water)^3 = volume of cone / volume of water
You can then find volume of water.
2007-12-30 3:58 am
Length of arc AB = 12*150/180*pi = 10pi (pi=3.14)
With arc AB, we can find r
2*pi*r = 10pi
r=5
CO^2 = 12^2-r^2
= 144-25
=119
CO = (119)^1/2
DO= ((119)^1/2)/2
Let x be the radius of the cone of water
By similarity of triangle, r/x = CO/DO
5/x =2
x= 2.5
Volume of water = 1/3* pi * x^2 *DO
= 1/3 * pi * 2.5^2 * ((119)^1/2)/2
= 35.68 cm^3
With arc AB, we can find r
2*pi*r = 10pi
r=5
CO^2 = 12^2-r^2
= 144-25
=119
CO = (119)^1/2
DO= ((119)^1/2)/2
Let x be the radius of the cone of water
By similarity of triangle, r/x = CO/DO
5/x =2
x= 2.5
Volume of water = 1/3* pi * x^2 *DO
= 1/3 * pi * 2.5^2 * ((119)^1/2)/2
= 35.68 cm^3
參考: 自己(我會考數學A)
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