Volume of a cylinder? V = πr²h. Rewritten as a diameter equation, this is:

V = π(d/2)²h = πd²h/4

Sub in h and V: 36p = πd²(4)/4 so 36p = πd²

Thus d = 6

LA = 2π(r)(h) = 2π(3)(5) = 30π

The equation for the volume of the cylinder is πr²h. When the radius doubles (r becomes 2r) you get π(2r)²h = 4πr²h. So when the radius doubles, the volume quadruples, giving a new volume of 80.

Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r. 

We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,

2πrh = 54π

Now we substitute 3r in for h.

2πr(3r) = 54π

6πr² = 54π

Divide by 6π

r² = 9.

Take the square root.

r = 3. 

h = 3r = 3(3) = 9.

Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr²h.

V = πr²h

V = π(3)²(9) = 81π

The answer is 81π.

The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πr_out² h – πr_in² h. The area of the cylinder using the outer radius is 80π cm³, and resulting hole is given by the volume from the inner radius, 20π cm³. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm³.

The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr².  Therefore, the total formula for the volume of the cylinder is: V = πr²h.

 First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.

Based on this, we know that the volume of our cylinder must be: π*12.5²*41.3 = 6453.125π in³

We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in³.

The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr²h = π * 2.5² * 8 = 50π in³

The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in³.

We must calculate our two volumes and subtract them. Following this, we will multiply by the density.

The volume of the cube is very simple: 12 * 12 * 12, or 1728 in³.

The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr²h = π * 3.75² * 12 = 168.75π in³.

The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in³. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.

The general form of our problem is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 5 * 6 * 10 = 300 in³

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is half the prism height, or 10/2 = 5 in. The area of the base is equal to πr². Note that the prompt has given the diameter. Therefore, the radius is 2, not 4. The base's area is: 2²π = 4π. The total volume is therefore: 4π * 5 = 20π in³.

The gel volume is therefore: 300 – 20π or (approx.) 237.17 in³.

We must find both the can volume and the gel volume. The formula for the gel volume is:

Gel volume = Prism volume – Can volume

The prism volume is simple: 12 * 13 * 42 = 6552 in³

The volume of the can is found by multiplying the area of the circular base by the height of the can. The height is one-fourth the prism height, or 42/4 = 10.5 in. The area of the base is equal to πr². Note that the prompt has given the diameter. Therefore, the radius is 4.5, not 9. The base's area is: 4.5²π = 20.25π. The total volume is therefore: 20.25π * 10.5 = 212.625π in³.

The gel volume is therefore: 6552 – 212.625π or (approx.) 5884.02 in³.

The approximate volume for the can is: 667.98 in³

From this, we can calculate the approximate mass of the contents:

Gel Mass = Gel Volume * 2.2 = 5884.02 * 2.2 = 12944.844 g

Can Mass = Can Volume * 1.5 = 667.98 * 1.5 = 1001.97 g

The total mass is therefore 12944.844 + 1001.97 = 13946.814 g, or approximately 13.95 kg.