A tricky volume of a cylinder math problem.

Let's find the volume of what we have on the board. Welcome back to my channel. So, this is not easy at all. Now, but we can do it together. First thing first, the volume of a cylinder. The volume of a cylinder is equal to area of the base, area of base multiplied by the height. Okay? Area of the base multiplied by the height. And our base is a circle, right? therefore the volume of this particular shape will be equal to area of this circle but the problem we have here is that we are not going to make use of the whole circle because the water in this tank is not full right and that is where we need to be careful how we go about it since the water is not full the area of this particular shape will be equal to the area of the the part that is filled with water right of the circle or of the tank multiplied by the height of the tank the height of the tank is 10 okay so this is where we are going to concentrate on because once we are able to get the area of this particular circle the shaded part of this circle we just multiply it by 10 to get our answer now this is the circle i want to draw it bigger and we have this this is the center of the circle and the radius of the circle is four but the height of the water is six the height of the water is six um what we are going to do here is basically to try to divide this circle into two shapes that we can actually find its area this is what i mean because from here to here is four it means that from here to here we also before the radius of the circle from here to here we also before the radius of the circle please subscribe to the channel turn on your notification bell if this is the first time you're coming across this channel hit the subscription button and give us a like if you actually love what we are doing thank you very much so at this point remember that from here to here is six right so it means that the height of our triangle will be height of the triangle will be equal to six minus four which is two so that means from here to here is two and remember that this is a chord right this particular line from the center to this place if we call this place a b a b is a chord right so whenever you draw a perpendicular from the center to the chord you have to get 90 degrees at this point of intersection and if that is so if we call the point of intersection n it simply means that let the center be O. It simply means that triangle OBN or ONB is a right angle, right? And if it's a right angle triangle, it's a right angled triangle. Now, if that is the case, that means that we can use the Pythagorean theorem to get NB, okay? So, this is what we have there, like this. This is 4, this is 2 this is n b okay remember that this is our drawing and as a result of that i will not be able to use this side of the board so we are going to manage what we have here therefore to get np is going to be nb squared let's call nbx so we have x squared plus two squared using pythagorean theorem we give us four squared okay x squared plus two squared we give us four squared Therefore, x squared will be 4 squared minus 2 squared. So x squared will be 4 squared is 16, 2 squared is 4. So x squared is 12. That means x is square root of 12. Okay, if x is square root of 12, remember that 12 can be written as 4 multiplied by 2. So we have square root of 4 multiplied by 2, which is, sorry, 4 multiplied by 3, rather. 4 multiplied by 3. And this should be equal to square root of 4 multiplied by square root of 3, which will give us 2 root 3. That means that NB is 2 root 3. So we have 2 root 3. But we need AB, the whole base of the triangle. So, this implies that AB is equal to 2 multiplied by 2 root 3. And this will give us 4 root 3. So, we have that our AB is 4 root 3, right? So, we are going to write it some way, that AB is 4 root 3. Alright, now why is this important? We need to find the area of this triangle, okay? So, to find the area of the triangle, okay, Now, area of the triangle is half base times height, right? So, we should be able to have multiplied by our base is AB. And AB is 4 root 3. So, we have 4 root 3. Multiplied by the height is just 2. So, if we solve this, we have that the area of the triangle is equal to 4 root 3 square units. 4 root 3 square units. So, we are going to write this somewhere. Area of triangle, 4 root 3. Okay. Now, remember that our intention is to find the area of this place, right? The shaded portion, which is from this triangle, this side. And you can see that this is what? This is a sector. This is a sector of a circle. This part is a sector of a circle. So, the area of the whole part that we are looking for, the area of this whole portion of the circle is going to be area of the triangle plus area of the sector okay area of the triangle plus area of the sector and we have found the area of the triangle and we need to find the area of the sector to find the area of the sector let's recall the formula for finding area of a sector okay now area of a sector is theta over 360 times pi r squared whereby theta is the angle here the angle we have here is our theta so let me clean this so the whole of this is our theta right so theta over 360 times pi r squared now let's also use this triangle to find the angle here so that we subtract it from 360 to get our theta so going back to our triangle let's pick half of it again so we have four here we have two here i remember that from here to here is two root three right two root three now let's get this let's call this alpha now this particular angle here can be gotten using our trigonometric ratio right so i want to make use of this side and this side this side is our adjacent this side is our hypotenuse, right? So, adjacent over hypotenuse is cos. So, we are going to have that cos alpha is equal to 2 divided by 4, which is half. And we know that cos alpha, that means that alpha is what? 60 degrees because cos 60 is half, which is 0.5. So, this implies that alpha is 60 degrees. So, if this place is 60 degrees, that means we have 60 here and we have another 60 here so all together here we have 120 right and if we have 120 here that means our theta will be 360 minus 120 which is 240 very important so that means this place is 240 degrees 240 degrees now that we have found the angle we can now use our formula to find the area of the sector so now let's solve the area of the sector area of the sector now will be equal to our theta is 240 divided by 360 multiplied by pi times r squared that is four squared so it will be equal to let's cancel zeros right and 24 what can go into 24 i'm going to take 6 and same time 6. 6 into 24 is 4 and 6 into 36 is 6. 2 into 4 is 2. 2 into 6 is 3. So now we have 2 over 3 times pi times 4 squared is 16. Now if we solve further, we are going to have this to be 2 over 3. Okay. Now 16 times 2 is 32. So we have 32 pi over 3. Now this is the area of the sector. This is the area of the sector. So, let's write it somewhere. So, area of sector is equal to 32 pi divided by 3. Now, remember that the area of this whole place now, all this shaded portion, right? The whole of this shaded portion from here is going to be equal to area of the triangle plus area of the sector. Okay? So, that will give us the area of the portion that contains the water. so let's go ahead and find it so this shaded portion now area of this shaded portion which is this will be area of the triangle which is 4 root 3 plus area of the area of the sector which is 32 pi over 3 now we are multiplying this by the height which is 10 so this will give us the volume of this tank right so the volume will be equal to now if we solve this you're going to have 40 root 3 if we open the bracket plus this will give us 320 pi divided by 3 so if we solve this further 40 root 3 if you press that in your calculator you're going to get 69.3 so we have 69.3 plus now 320 pi divided by 3 we give us 335.1 now some will be saying how did this man get this i've already punched that in my calculator i'm not an alien okay so this is what you're going to get and then you add these both if you add both you're going to get 404.4 square units oh sorry volume cube units okay cubed now this is what you're going to get as the answer 404.4 um units cubed so thank you so much for watching and see you in my next video bye