Video: CBSE Class X • Pack 4 • 2015 • Question 31

CBSE Class X • Pack 4 • 2015 • Question 31

04:44

Video Transcript

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 sevenths.

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 this formula.

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 14.

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 centimeters.

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