Video: CBSE Class X • Pack 5 • 2014 • Question 34

CBSE Class X • Pack 5 • 2014 • Question 34

05:16

Video Transcript

A solid cylinder has a height of 2.8 centimeters and a diameter of 4.2 centimeters. If a conical cavity of the same height and diameter is hollowed out of the cylinder, what is the total surface area of the remaining solid? Take 𝜋 is equal to 22 over seven.

Let’s start by drawing a diagram of the solid we’re finding the surface area of. And we can label on the height of the solid and its diameter. Now, let’s consider the surface area of this shape.

We have that the total surface area will be equal to the surface area of this cylinder minus the surface area of the circular face at the top plus the surface area of the cone minus the area of the circular face of the cone, which is the same circular face missing from the cylinder.

Now, let’s recall some equations. We have that the surface area of a cylinder is equal to two 𝜋𝑟ℎ plus two 𝜋𝑟 squared. The surface area of a cone is equal to 𝜋𝑟 squared plus 𝜋𝑟 multiplied by the square root of 𝑟 squared plus ℎ squared. And the area of a circle is equal to 𝜋𝑟 squared. So let’s substitute these into our equation for the total surface area.

So the surface area of the cylinder is 𝜋𝑟ℎ plus two 𝜋𝑟 squared. The area of the circular face is 𝜋𝑟 squared. The surface area of the cone which is 𝜋𝑟 squared plus 𝜋𝑟 multiplied by the square root of 𝑟 squared plus ℎ squared. And then, finally, the area of the lower circular face which is again 𝜋𝑟 squared.

Now, we know that 𝑟 and ℎ are gonna be the same for the cylinder and the cone since the question tells us that the conical cavity has the same height and diameter as the cylinder. And we know that the radius is equal to half the diameter. And so the radius of both the cone and the cylinder is 2.1 centimeters. Now, let’s simplify our equation.

We notice that we have many 𝜋𝑟 squared terms. And if we group these together, what we’re left with is simply 𝜋𝑟 squared. And this leaves us with the equation that the total surface area is equal to 𝜋𝑟 squared plus two 𝜋𝑟ℎ plus 𝜋𝑟 times the square root of 𝑟 squared plus ℎ squared. And to neaten this up a little bit, we can factorize out 𝜋𝑟. And so this is the simplified version of our equation for the total surface area.

Now, let’s substitute in our values for the radius 𝑟 which is 2.1 centimeters and the height ℎ which is 2.8 centimeters. And this gives us 𝜋 times 2.1 times 2.1 plus two times 2.8 plus the square root of 2.1 squared plus 2.8 squared. Now, this all looks quite straightforward to simplify apart from the square root at the end. So let’s focus on simplifying this first.

In order to make this a little bit easier for ourselves, let’s write these decimals as fractions. And so we have that our square root is equal to the square root of 21 over 10 squared plus 28 over 10 squared. This is equal to 21 squared plus 28 squared all over 10 squared. And now, we know that 21 and 28 are both multiples of seven. And so we can write this as seven times three all squared plus seven times four all squared over 10 squared.

And we can factorize out the seven squared over 10 squared. And we’re left with the square root of seven squared over 10 squared multiplied by three squared plus four squared. And now, we can take the square root of seven squared over 10 squared to get seven over 10. And we still have to multiply this by the square root of three squared plus four squared. And now, we have that three squared is simply nine and four squared is 16.

And so we have seven over 10 times the square root of nine plus 16. But nine plus 16 is simply 25. And we’re left with seven over 10 multiplied by the square root of 25. And since the square root of 25 is five, we’re left with seven over 10 multiplied by five. And this is simply equal to seven over two or 3.5.

And now, we found the value of the square root of 2.1 squared plus 2.8 squared to be 3.5. And we can substitute this back into our equation for the total surface area. And so now we’re ready to calculate the surface area. We can multiply the 2.8 by two to get 5.6. And then, adding the numbers together in the bracket gives us that the total surface area is equal to 𝜋 times 2.1 times 11.2.

And now, we can use the value of 𝜋 given in the question. So that’s 22 over seven. And this becomes 22 over seven times 2.1 times 11.2. And now, we know that we can write 2.1 as seven multiplied by 0.3. And this becomes equal to 22 over seven times seven times 0.3 times 11.2. And now, we are dividing and multiplying by seven. So these two cancel out.

And what we’re left with is 22 times 0.3 times 11.2. And 22 times 0.3 is simply 6.6. And this becomes 6.6 multiplied by 11.2. And we can work out 6.6 times 11.2 to be 73.92. And so we found that the total surface area of the solid is 73.92 centimeters squared.

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