
A test-tube consists of a hemisphere and a cylinder of the same radius. The volume of water required to fill the whole tube is and of water are required to fill the tube to a level which is 2 below the top of the tube. Find the radius and volume of the whole tube.
Answer
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Hint: First we consider a test tube of a hemisphere and a cylinder of the same radius.
We use the formula for volume of the hemisphere and volume of the cylinder then we add these volumes and find the volume of a test tube.
Finally we subtract both the volume to get the radius of these test tube
Complete step-by-step answer:
It is given that the volume of water required to fill the entire tube is ,
the volume of water-filled upto 2cm is
Let be the radius of the test tube and
Let be the height of the test tube.
Volume of the test tube = volume of the hemisphere + volume of the cylinder
=
= .........(1)
water is filled below the top of tube,
Now, volume
=
= ..........(2)
Subtracting the equation we get,
Putting and dividing the values we get
Taking root on both sides we get,
Substituting the value of r in equation (1) we get,
which implies that,
Divide 67 by 3 we get the value of height
Therefore, the radius of the test tube is and the height
Note: Students make common mistakes i.e. they have not considered the marked point which is from the top of the tube.
We use the formula for volume of the hemisphere and volume of the cylinder then we add these volumes and find the volume of a test tube.
Finally we subtract both the volume to get the radius of these test tube
Complete step-by-step answer:
It is given that the volume of water required to fill the entire tube is
the volume of water-filled upto 2cm is
Let
Let
Volume of the test tube = volume of the hemisphere + volume of the cylinder
water is filled
Now, volume
Subtracting the equation
Putting
Taking root on both sides we get,
Substituting the value of r in equation (1) we get,
which implies that,
Divide 67 by 3 we get the value of height
Therefore, the radius of the test tube is
Note: Students make common mistakes i.e. they have not considered the marked point which is
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