Video: CBSE Class X • Pack 4 • 2015 • Question 17

CBSE Class X • Pack 4 • 2015 • Question 17

05:57

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.

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