A copper wire of dia 6mm is evenly wrapped around a cylinder of length 18cm & dia 49 cm to cover the whole surface. Find volume and length of wire used.
To figure out how much wire you'll be using, you just recognize that each section of wire can cover 6 mm of height of the cylinder. This means that to figure out how many loops of wire (n) you need, you just divide the height (length) of the cylinder by 6 mm (because the length of the cylinder is in cm, we'll convert 6 mm to 0.6 cm)
n = 18 cm / 0.6 cm = 30 loops
You also need to recognize that the circumference for the wire wrapped around the cylinder will be slightly larger than the circumference of the cylinder. This is because the diameter of the wire is 6 mm. If you can imagine, the average circumference will be based on the center of the wire. This is demonstrated in the graph below. The inner circle is the cylinder, and the outer circle is the wire. You could imagine the average circumference goes between the two red dots (we need to use average circumference because the length of the wire is based on the center of the wire).
When you consider that in a cross-section of the cylinder + wire that you will be taking the diameter of the cylinder and add TWO radii of the wire (keep in mind, the radius is just half of the diameter = 3 mm = 0.3 cm), you get the following for the length of wire in one loop (l) using the formula for circumference (C = pi*diameter):
`l = pi*(d_(cyl) + 2*r_(wire))`
`l = pi*(49 + 2*(0.3))=155.8 cm`
Now, we know how many loops we need and the length of each loop. To figure out the total length of wire (L), we just multiply these two numbers!
`L = l*n`
`L = 155.8 * 30 = 4674`
So we have one of our answers: the total length of wire is 4674 cm.
Now, we just need to find the volume. Recall, the wire can be considered just a long, thin cylinder. The formula for the volume of a cylinder can be expressed in terms of the length and the radius of a cylinder (see link below):
`V_(cyl) = pir^2h`
Using our wire radius (keep in mind, since everything is in cm, we'll be using 0.3 cm for the radius again) and our total length:
`V_(wire) = pi(0.3)^2(4674)`
`V_(wire) = 1321.5`
And there you have it! The volume of our wire is 1321.5 `cm^3`
Hope that helps!