Find the rate of change of the volume of a cylinder when its radius is 6 feet if its height is always (3/2) times its radius and its radius is increasing at the rate of 2 feet per minute.

Assuming you are correct that the radius is 2' 6" and not the diameter.
Convert the inches to centimetres
1" = 2.54 Cms
Radius = 2' 6" = 30" X 2.54 = 76.2 Cms
Height = 5' 2" = 62" X 2.54 = 157.48 Cms
Volume of cylinder = Pi x radius squared x height
3.142 x 76.2 x 76.2 x 157.48 = 2,873,038.9 cc
Divide by 1000 to find how many litres (interest note 1000cc car engine is 1 litre)
2,873,039 cc / 1000 = 2,873 litres

The formula is V = Pi * R^2 * H where

R: radius
H: height
V: volume

So you need the radius and height in feet, and plug them into this formula. Or you can enter what you know, in feet, to this calculator, and it will give the unknown

http://www.onlineconversion.com/object_volume_cylinder_tank.htm
.

Hi Paula:
A mathematical logic question.
You need to get all of the measurements in the same units.
- A cubic yard is 1 yard x 1 yard x 1 yard
- equals 3 ft x 3 ft x 3 ft
- equals 27 cubic feet,
so once you have the volume of the pipe in cubic feet, you divide by 27 to get the volume in cubic yards.
- Volume of a cylinder (the pipe) is the cross sectional area X length.
- Area equals Pi x r squared.
- equals 22/7 x 1 x 1
equals 3.14#####
times length (20)
equals 62.857### cubic feet volume
divide by 27 cubic feet
equals 2.328### cubic yards.

- The answer is always the easy part.
- what is important is knowing exactly what the question is, and
- determing what calculations are needed to get the answer.

Wasn't that better than if I'd just said 2 1/3 cubic yards?
Cheers

The radius or the volume of the space?

Volume of right cylinder is
V=(Area of base)*height.
If cylinder is circular Area of base =PI*radius^2=Pi*r^2
Volume is V=Pi*(8^2)*6 cm^2
Finish the calculation.

Volume of right cylinder is
V=(Area of base)*height.
If cylinder is circular Area of base =PI*radius^2=Pi*r^2
Volume is V=Pi*(8^2)*6 cm^2
Finish the calculation.

Volume of a right circular cylinder of radius r and height h is
V=PI*(r^2)*h
Figure out which is the radius and which is the height, then plug in the formula above. Result is in cm^3

volume is calculated by finding the area multiplied by the length or height . That is Pi X radius squared X length
example --tube 6"dia 12"long what is the volume?
Example 3.1417 X radius 3" Squared =3.1417X9sq " = 28.2740sq" X length 12" =338.688 cubic"
If you want the volume of an engine cylinder , it is the same formula and the length is the stroke of the piston ( distance from TDC to BDC )

You need a cone with a volume of 150 gallons.
The volume of a right circular cone is given by the formula
V_cone=(Area of base *height)/3
Since cone is circular , its base is a disk with a certain radius r. The formula becomes PI*(r^2)*h/3
As you see there are two variables, the radius and the height. There is an infinite number of combinations (radius, height) that will give you a total volume of 150 gallons. You can choose the radius (r) and solve for h, or you can choose the height then solve for the radius.
I suggest you convert gallons to cubic feet or to cubic meters then put
V=Pi*(r^2)*h/3 where V is the value of volume after conversion.
If you choose the radius, then to isolate h, h=(3*V)/(Pi*r^2)
If you choose the height, then to solve for r
r^2=(3V)/(Pi*h)
and r=square root of (3V/(Pi*h))
Now, it is your turn to carry out the rest of the calculation.
It will consist of the value of the radius, and the value of the height.
If the cone is not right circular, the situation becomes more complicated.

A pipe can be seen as two cylinders, one of "air" sitting inside the other, in this case lead.
To get the volume of a pipe you thus subtract the volume of the "air" cylinder from that of the lead cylinder.
Volume of a cylinder = area of circular cross section (or pi times square of the radius) times the height (or length) => ? * r2 * h
Volume of a pipe = Volume of cylinder 1 (outer diameter) - volume of cylinder 2 (inner diameter).

In this case; Volume of outer cylinder => inner diameter 1/2" (0.5") + pipe thickness 1/4" (0.25") = 3/4" (0.75") divided by 2 to get radius = 0.375".
3.14 (?) * 0.375 * 0.375 * 120(10 feet) = 53.01438 in3

Volume of inner cylinder => inner diameter 1/2" (0.5") divided by 2 to get radius = 0.25".
3.14 (?) * 0.25 * 0.25 * 120(10 feet) = 23.56194 in3

Volume of lead in this pipe = 53.01438 - 23.56194 = 29.45244 in3

Hope this helps ;)