### Video Transcript

Due to sudden floods, some welfare
associations jointly requested that the government set up 100 tents immediately. And the associations offered to
contribute 50 percent of the cost. The lower part of each tent is
cylindrical with diameter 4.2 metres and height four metres, whereas the upper part
is conical with the same diameter but with height 2.8 metres. If the canvas used costs 100 rupees
per metre squared, find the amount paid by the associations. Use 𝜋 is equal to twenty-two
sevenths.

For this question, we’ll need to
find the total surface area of the tent. Since it’s made up of two parts, we
should calculate the curved surface area of the cylinder and the curved surface area
of the cone. The formula for curved surface area
of a cone is 𝜋𝑟𝑙, where 𝑙 is the slanted height of the cone and 𝑟 is the
radius.

Let’s look at a side view of the
cone. It has a vertical height of 2.8
metres and a diameter of 4.2 metres. We know that the radius can be
calculated by halving the diameter. That’s 2.1 metres.

The length of the slanted height
requires a little more work. We know the height of the cone is
2.8 metres, and the vertical height and the radius meet at an angle of 90
degrees. So we can use the Pythagorean
theorem to find the length of the slanted side.

Remember, the Pythagorean theorem
says that the square of the longest side is equal to the sum of the squares of the
two shorter sides. That’s sometimes written as 𝑎
squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the hypotenuse of the
triangle. Substituting the lengths from our
triangle into this formula, we get 2.1 squared plus 2.8 squared is equal to 𝑐
squared.

Now we could evaluate 2.1 squared
and 2.8 squared, find their sum, and then square-root it to find the length of
𝑐. However, 2.1 squared and 2.8
squared are two parts of a Pythagorean triple. In fact, 21 squared plus 28 squared
is equal to 35 squared. We can scale this down to say 2.1
squared plus 2.8 squared is equal to 3.5 squared. And in doing so, we can see that
the length of the hypotenuse of our triangle is 3.5 metres. The slanted height of the cone is
3.5 metres.

Now that we know the slanted
height, we can substitute what we know into the formula for the curved surface area
of the cone. It’s 𝜋 multiplied by 2.1
multiplied by 3.5. We’ve been told to use 𝜋 is equal
to twenty-two sevenths. And to make this formula a little
bit more manageable, we can replace 2.1 with twenty-one tenths and 3.5 with seven
over two. We can then cross-cancel by
dividing through by seven and by two. And our calculation becomes 11
multiplied by 21 all over 10. 10 multiplied by 21 is 210, so 11
multiplied by 21 is 210 plus another 21. It’s 231. The surface area of the cone is
given by two hundred and thirty-one tenths, which is equivalent to 23.1 metres
squared.

Next, we need to calculate the
curved surface area of the cylinder. That’s given by the formula two
𝜋𝑟 multiplied by ℎ, where 𝑟 is the radius of the cylinder and ℎ is its vertical
height. Once again, the radius of the
cylinder is half of its diameter. It’s 2.1 metres. And the height of the cylinder is
four metres. So our formula becomes two
multiplied by 𝜋 multiplied by 2.1 multiplied by four.

Once again, we can replace 𝜋 with
twenty-two sevenths and rewrite 2.1 as twenty-one tenths. We can then cross-cancel by
dividing through by seven and then by two. That leaves us that the curved
surface area of the cylinder is 22 multiplied by three multiplied by four all
divided by five. 22 multiplied by three is 66, and
we can then multiply this by four by timesing by two and then timesing by two
again. 66 multiplied by two is 132, and
timesing that by two again, we get a value of 264. We’ve calculated the curved surface
area of the cylinder to be 264 over five metres squared.

Now at this stage, we could use the
bus stop method to evaluate this. However, if we multiply both the
numerator and the denominator of this fraction by two, we get that it’s equivalent
to five hundred and twenty-eight tenths, which is equal to 52.8 metres squared. The curved surface area of the
cylinder is 52.8 metres squared. Adding these two numbers together,
and we get that the total surface area of one tent — that’s the total amount of
canvas required — is 75.9 metres squared.

There are 100 tents, so we can
multiply this number by 100. And that tells us that the total
amount of canvas required is 7590 metres squared. Since each metre squared of canvas
costs 100 rupees, multiplying the total amount of canvas required by 100 gives us a
total cost of 759000 rupees. Finally, we’re told that the
associations offered to contribute 50 percent of the cost. We can find 50 percent of a number
by halving. Half of 759000 is 379500. The total cost for the associations
is 379500 rupees.