A hemisphere is scooped out from
each end of a solid metal cylinder such that both hemispheres are of the same
diameter as the cylinder. The height of the cylinder is 10
centimeters and the radius of its base is 4.2 centimeters. If the rest of the cylinder is
melted and converted into a cylindrical wire with a thickness of 1.4 centimeters,
find its length. Use 𝜋 is equal to twenty-two
We are given that when the two
hemispheres are removed, the rest of the cylinder is melted and converted into a
cylindrical wire. Assuming that no metal is lost we,
can say that the volume of the wire is going to be equal to the volume of the
cylinder minus the volume of the two hemispheres.
We should, therefore, begin by
deriving an expression for the volume of the cylinder and the two hemispheres. The formula for volume of a
cylinder is 𝜋𝑟 squared ℎ. And the formula for the volume of
the two hemispheres is simply the formula for the volume of a sphere; that’s
four-thirds 𝜋𝑟 cubed.
Now, we’re told that the
hemispheres and the cylinder have the same diameter. This means they have the same
radius. And we can use the same radius
throughout this question. Let’s substitute what we know into
The radius of the base of the
cylinder is 4.2. So the volume of the cylinder is 𝜋
multiplied by 4.2 squared multiplied by its height which is 10. The volume of the two hemispheres
which we can assume is equivalent to the volume of one sphere is four-thirds
multiplied by 𝜋 multiplied by 4.2 cubed.
Now, at this stage, we would
usually substitute 𝜋 for twenty-two sevenths. However, at this time, we’re going
to leave our answer in terms of 𝜋. This expression does look quite
complicated. But notice there’s a common factor
of 𝜋 and 4.2 squared in each part. We can, therefore, factorize by
taking out 4.2 squared 𝜋. And that gives us 4.2 squared 𝜋
multiplied by 10 minus four-thirds of 4.2. Four-thirds of 4.2 is 5.6.
We can perform a little bit of
cancelling by dividing through by three. So our expression for the volume of
the wire is 4.2 squared 𝜋 multiplied by 10 minus 5.6. 10 minus 5.6 is 4.4. And therefore, the volume of the
wire is given by 4.2 squared 𝜋 multiplied by 4.4.
Now, we’re actually told that the
wire is cylindrical. So we can once again use the
formula for volume of a cylinder to form an expression for the volume of wire in
terms of its length. The wire has a thickness of 1.4
centimeters; that’s its diameter. If we halve this, we can see that
the radius of the wire is 0.7 centimeters.
Substituting what we know about the
cylindrical wire into the formula for volume of a cylinder and we get 𝜋 multiplied
by 0.7 squared multiplied by ℎ, which we’ll call the length of the wire.
You might now be able to see why we
left it in terms of 𝜋. We can divide everything through by
𝜋. And we’re left with 0.7 squared ℎ
is equal to 4.2 squared multiplied by 4.4. Next, we’ll divide everything
through by 0.7 squared. This might still look a little
complicated. However, we can rewrite 4.2 squared
over 0.7 squared as 4.2 over 0.7 all squared. 4.2 divided by 0.7 is six. So the length of our wire ℎ is
given by six squared multiplied by 4.4.
Six squared is 36. So all we need to do to find the
length of the wire is multiply 36 by 4.4. In fact, we’ll multiply 36 and 44
first. We can use a formal written method
for multiplication such as the column method. Six multiplied by four is 24. Three multiplied by four is 12. And when we add the four, we get
Next, we’re going to multiply 36 by
this other four. That’s in fact multiplying by
40. So we add a zero in this column to
show that everything is going to be 10 times bigger than the number we get. Once again, six multiplied by four
is 24 and three multiplied by four is 12. Adding the two, we get 14. Adding these two numbers and we get
1584. 4.4 is 10 times smaller than
44. So our answer is also going to be
10 times smaller.
The length of the wire is 158.4