The dimensions of a solid iron
cuboid are 4.4 metres by 2.6 metres by one metre. It is melted and recast into a
hollow cylindrical pipe of inner radius 30 centimetres and thickness five
centimetres. Find the length of the pipe.
In this scenario, we have two
different shapes. First, we start out with a cuboid
with given dimensions, which we’re told is made of metal and melted down into a
totally different shape — a hollow cylinder. The hollow cylinder has an inner
radius — we’ve called 𝑟 sub 𝑖 — which is given as 30 centimetres and a wall
thickness of five centimetres, which we’ve called 𝑡.
Knowing all this, we want to solve
for the length of the hollow cylinder 𝑙. If we label the volume of the
cuboid 𝑉 sub 𝑐, we’re told that volume is 4.4 times 2.6 times one cubic
metres. When it comes to the volume of the
hollow cylinder, we can start calculating that by recalling that the area of a
circle is equal to 𝜋 times the radius of this circle squared.
On the end of the cylinder, we want
to solve for the shaded area, which is the difference between the area of two
circles. If we call the volume of the hollow
cylinder 𝑉 sub 𝑝 for the volume of the pipe, that volume is equal to the area of
the larger circle on the end of the pipe 𝜋 times the inner radius plus the pipe
thickness all squared minus the area of the smaller hollow circle on the end of the
pipe 𝜋 times the inner radius squared all multiplied by 𝑙, the length.
We know that the volume of the
cuboid is equal to the volume of the pipe. So we can say that the volume of
the cuboid is equal to 𝜋 times the radius of the outer circle squared minus the
radius of the inner circle squared all multiplied by the length of the pipe 𝑙.
Looking at the different units in
this expression, we see that on the left-hand side we have length units of metres,
while on the right we have length units of centimetres. Recalling that one metre is equal
to 100 centimetres, we can rewrite our left-hand side as 440 times 260 times 100
cubic centimetres. And our right-hand side simplifies
to 𝜋 times 35 squared square centimetres minus 30 squared square centimetres all
multiplied by 𝑙. If we multiply 35 by itself, we
find a result of 1225. And then, when we square 30, we
find a result of 900.
When we then subtract the area of
the inner circle from the area of the outer circle on the end of the pipe, we find a
result of 325 square centimetres. Then, if we let the number 𝜋 equal
exactly 3.14, we can multiply that value by the number of square centimetres on the
right-hand side of our equation. When we do, we find a result of
And next, to convert this into an
integer, we can multiply both sides of our equation by two. This leaves us with the equation
880 times 260 times 100 cubic centimetres is equal to 2041 square centimetres
multiplied by 𝑙. We can rearrange the numbers on the
left-hand side of our equation so that it reads 88 times 26 times 10000. Then, when we multiply 88 by 26, we
find a result of 2288.
With that number on the left-hand
side of our expression, we’re now ready to solve for the length 𝑙. To do that, we’ll divide both sides
of our equation by 2041 square centimetres. This cancels this term out on the
right-hand side. And considering the units on the
left-hand side, we see that two factors of centimetres cancel out, leaving us with
units of centimetres for our length.
We’re now ready to calculate the
pipe length 𝑙. And we’ll do it by dividing 2041
into 22880000. To start off, 2041 goes into 2288
one time. Then, it divides into 2470 one
time, it divides into 4290 two times, into 2080 one time, and finally into 390 zero
This means that 𝑙 is equal to
11210 centimetres or 112.10 metres. That’s the length of the pipe.