Can someone help me with this question:
A cylindrical tin has an internal diameter of 18cm. It contains water to a height of 13.2cm. When a heavy spherical ball of diameter 9.3cm is immersed in it, what is the new height of the water level?
And if you could explain how you did it that would be great, thanks :)
A cylindrical tin has an internal diameter of 18cm. It contains water to a height of 13.2cm. When a heavy spherical ball of diameter 9.3cm is immersed in it, what is the new height of the water level?
And if you could explain how you did it that would be great, thanks :)
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internal diameter of cylinder = 18cm so radius = 9 cm
height of water = 13.2 cm
volume of water = ( pir^2 h) = pi* 9^2* 13.2= 1069.2 pi cm^3
volume of ball = 4/3 pi (9.3/2)^3 =4/3pi x 100.5446
=134.0595 pi cm^3
total volume = volume of water + volume of ball =( 1069.2+ 134.0595)pi cm^3
= 1203.2595 pi cm^3--------------------(I)
now let the height of water rises to h cm
so total height = ( 13.2 + h)
so new volume = pi9^2 ( 13.2 + h)---------------------------(II)
so (I) = (II)
pix81 (13.2 + h) =1203.2595 pi
13.2 + h = 1203.2595/81 = 14.855
so new height = 14. 855 cm
---------------------------------.,-----… alternately----------
pi 9^2 h = 134pi
or h = 134/81 = 1.654
so new height = 13.2 + 1.654 = 14. 854
height of water = 13.2 cm
volume of water = ( pir^2 h) = pi* 9^2* 13.2= 1069.2 pi cm^3
volume of ball = 4/3 pi (9.3/2)^3 =4/3pi x 100.5446
=134.0595 pi cm^3
total volume = volume of water + volume of ball =( 1069.2+ 134.0595)pi cm^3
= 1203.2595 pi cm^3--------------------(I)
now let the height of water rises to h cm
so total height = ( 13.2 + h)
so new volume = pi9^2 ( 13.2 + h)---------------------------(II)
so (I) = (II)
pix81 (13.2 + h) =1203.2595 pi
13.2 + h = 1203.2595/81 = 14.855
so new height = 14. 855 cm
---------------------------------.,-----… alternately----------
pi 9^2 h = 134pi
or h = 134/81 = 1.654
so new height = 13.2 + 1.654 = 14. 854
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start by finding the volume of water in the tin and also the volume of the sphere, then add these volumes together and you get the new volume inside the tin. Then solve for the height of a cylinder with that new volume and with a diameter of 18cm.
Here's the math
Volume in tin = pi*18^2*13.2 = 4276.8*pi cm^3
Volume of sphere = (4/3)*pi*9.3^3 = 1072.476*pi cm^3
Sum of these volumes = 5349.276*pi cm^3
height of cylinder with given volume = (5349.276*pi cm^3) / (pi*18^2 cm^2) = 16.51 cm
Here's the math
Volume in tin = pi*18^2*13.2 = 4276.8*pi cm^3
Volume of sphere = (4/3)*pi*9.3^3 = 1072.476*pi cm^3
Sum of these volumes = 5349.276*pi cm^3
height of cylinder with given volume = (5349.276*pi cm^3) / (pi*18^2 cm^2) = 16.51 cm
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The ball displaces an equal volume of water. So, when you immerse the ball, you first determine the volume of the ball. the volume of the ball is 4/3*pi*(d/2)^3= 4/3*pi*(9.3/2)^3=(9.3)^3*pi/6=421.16 cm^3. Now in a cylindrical tin, the amount of water from the bottom to a height of 1 cm is
pi(18/2)^2(1)= 81pi= 254.47.
So now form a ratio 1 cm is to 254.47 as x cm is to 421.16.
Solves 1/254.47=x/421.16, so x= 421.16/254.47= 1.66 cm.
So the new height is 13.2 cm + 1.66 cm= 14.86 cm
This works because a cylinder is symmetrical as you go up. If you were in a cone, you couldn't do it that way. A direct way is to solve pir^2h= 421.16 for h because you know r.
pi(18/2)^2h=421.16
h=1.66 cm so the height of the water is now 14.86 cm
pi(18/2)^2(1)= 81pi= 254.47.
So now form a ratio 1 cm is to 254.47 as x cm is to 421.16.
Solves 1/254.47=x/421.16, so x= 421.16/254.47= 1.66 cm.
So the new height is 13.2 cm + 1.66 cm= 14.86 cm
This works because a cylinder is symmetrical as you go up. If you were in a cone, you couldn't do it that way. A direct way is to solve pir^2h= 421.16 for h because you know r.
pi(18/2)^2h=421.16
h=1.66 cm so the height of the water is now 14.86 cm
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Volume of the ball is (4/3)pi * (r^3) = 421.2 cm^3.
Volume of tin per unit height = pi(r^2) x 1 cm (height) = 254.5 cm^3.
So the rise in height = 421.2 cm^3 / 254.5 cm^3 = 1.655 cm.
Final height is 13.2 + 1.7 = 14.9 cm
Volume of tin per unit height = pi(r^2) x 1 cm (height) = 254.5 cm^3.
So the rise in height = 421.2 cm^3 / 254.5 cm^3 = 1.655 cm.
Final height is 13.2 + 1.7 = 14.9 cm