### 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.