Let's do this question, a conical Wesson is full of water, a spears dropped into the vessel such that
it touches the sides and is just immersed in the radius of the clinical vessel is nine centimeter and
height is 12 centimeter.
What fraction of water spills out?
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Now, over here we have a conical vessel and a spear is dropped.
Right.
And it's given that it touches it and it is just immersed.
So this is how the spear is located.
Now we need to find the fraction of water that spills out.
Right.
What does that mean?
That would be the volume of the spear.
Right.
Divided by the volume of the cone, because the fraction of water that spills out will be equal to the
volume of the spear.
Right.
And because we want to know what percentage fraction of water that spills out means, what percentage
of the water that's already there in the cone is going out.
So that's why we need to take the ratio volume of the spear divided by volume of the cone.
All right.
Well, let me draw a line over here.
OK, now it's given that the radius is nine centimeter and the height is 12 centimeter to over here.
If you take this part, you see you get it right angled triangle.
Let me draw that over here.
So this is nine centimeters.
This is 12 centimeters.
So Asper Pythagoras's theorem, this side will be root of 12 squared plus nine square.
That's equal to 15.
Right.
So this side over here is 15.
All right.
Now, let me draw this part over here.
All right, so I have nine over here and nine over here because the radius is given us nine.
Right.
And this spear is touching over here because it is just immersed and it touches the core.
Right.
So you have a point of contact over here and the point of contact here.
Now, you can see that this and this has to be equal, right?
Because when you look at it from this section, this is actually a tangent from an external point to
a circle right there for this.
And this has to be equal to this and this has to be equal.
Let's make sure that this is nine centimeter and this is nine centimeter.
All right.
Now, over here, we know that this side over here, that is this much has to be 15 centimeters, therefore,
because this part over here is nine.
This part over here has to be 15 minus nine.
That's equal to six centimeters.
Right.
And similarly, over here also you have six centimeters.
Now, this circle over here is the inner circle of this triangle.
Right?
So let's mark that over here.
This would be the in radius.
All right.
And remember, so because this is inscribed inside this triangle, these angles will be 90 degrees.
Right.
Or these.
Sites are tangents, all right, now over here you have three areas are and we know that the high desert
will there.
Therefore this measurement will be 12 miners are right now.
Let's take out this triangle over here where this site is.
12 miners are.
One are.
And this site is six.
Let's draw it over here.
So you have a triangle over here where this site is 12 miners are.
This is odd.
And this is six right now.
As per Pythagoras's theorem, 12 miners are square will be equal to our square plus six square.
That's 36.
So let's simplify this.
We get one forty four plus square minus 24 hour is equal to our square plus 36 or on simplifying, you
get twenty four hours equal to one zero eight.
And that gives you are is equal to four point five.
Right now we need to find the ratio of the volume of the spill to the volume of the cone.
Right.
But let me make some space over here.
We know that the volume of the spear will be four by three by tube and the volume of the cone is one
by three by a square inch.
Right now, in these two cases, the radius is not the same.
Right over here we are taking the radius of the spear and this is the radius of the cone.
Now, let's substitute the values over here.
And remember, we can cut out by and by and three and three right to all your radius of the spear is
equal to four point five centimeter and radius of the cone we have about nine centimeter and height
is equal to 12 centimeters.
So let's substitute we get four into four point five.
The whole cube divided by nine square into 12.
Right.
And this becomes nine by twenty four.
And that is your answer, right.
You can simplify it to three by eight.