Volumes without calculus Solve the following problems with and...

Question

Volumes without calculus Solve the following problems with and without calculus. A good picture helps.

b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a cube is inscribed in a right circular cylinder with a base radius of 2 units and a height of 4 units, such that its base sits on the cylinder's base and its top touches the cylinder's side. What is the volume of the cube? So we're going to begin by drawing a diagram of our setup. So we're going to be looking at our system from above. So if we look down onto our cylinder, we're just going to see a circle. So we'll draw a circle to represent our cylinder, and our cube needs to be inscribed inside the cylinder. So if I look down at my cube from above, I'm just going to see a square. So we're going to inscribe a square inside of our circle. Now we're told that the top of our cube touches the cylinder's side. So that means that the top base of our cube is going to be touching the side of the cylinder at its four corners. And so in our diagram here we see that our square here is touching the circle at its 4 corners. We're also told that the base of our cube sits on the cylinder's base. And so that's going to allow us to find the diagonal of the square that we have drawn in our figure. So the diagonal of the square that we've drawn in our figure is equal to the diameter of our circle. We're told that the base radius of our cylinder is 2 units, and so we'll need to double that to find the diameter. So the diameter of our circle here, as well as the diagonal of our square, is going to be 4 units. And what we want to do is relate this diagonal to the side of one of our squares, which we label as S, which is also going to be the side of our cube. And if we can find S here, that means that we can use it to find the volume. Now recall the relationship between the diagonal of a square and the length of its side. So that is the diagonal, which in our case is 4, is going to be equal to S multiplied by the square root of 2. So, now we need to solve this equation for S. To do that, we'll just divide both sides of the equation by the square root of 2, and we see that S is going to be equal to 2 multiplied by the square root of 2. So now we need to need to use this to find the volume and recall that our volume of a cube V is going to be equal to Scubed. So this is going to be equal to the quantity of 2 multiplied by the square root of 2 in quantity, cubed. And evaluate in this expression, we see that our volume is going to be equal to 16 multiplied by the square root of 2 cubic units. But this here is the answer that we're looking for for this problem. And so I hope that this explanation has been useful, and I'll see you in the next video.

Volumes without calculus Solve the following problems with and without calculus. A good picture helps.b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?

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Volumes without calculus Solve the following problems with and without calculus. A good picture helps.

b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?