SOLUTION: From a cylindrical object of diameter 70cm and height 84cm, a right solid cone having its base as one of the circular ends of the cylinder and height 84cm is removed. Calculate:

Question 1205078: From a cylindrical object of diameter 70cm and height 84cm, a right solid cone having its base as one of the circular ends of the cylinder and height 84cm is removed.
Calculate:
a) The volume of the remaining solid object, expressing your answer in the form of a × 10ⁿ where 1 < a < 10 and n is a positive integer.
b) The surface area of the remaining solid object
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (a)

If a cylinder and cone have the same radius and height, then we have this very interesting connection:
cone = (1/3)*cylinder
or
cylinder = 3*cone

This is an informal way of saying "we need 3 cones to make a cylinder".
In other words, 3 cone volumes combine to a cylinder volume.
Again, both must share the same radius and same height.

Luckily the radius values are the same because we're carving a cone out of the cylinder.
And both heights are the same as well (84).

After carving a cone out of the cylinder, the cylinder loses 1/3 of its volume and keeps the remaining 2/3.

volume of cylinder = pi*r^2*h
volume of cylinder = pi*(70/2)^2*84
volume of cylinder = 102900pi

2/3 of that volume is (2/3)*102900pi = 68600pi which is the leftover amount.

Your teacher hasn't stated something like "use pi = 3.14", so I'll use the calculator's stored version of pi instead.
68600pi = 215,513.25603626
which rounds to 215,513

That converts to the scientific notation 2.15513 * 10^5
This is because we move the decimal point 5 spots to the right to go from 2.15513 back to 215,513 again.

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Part (b)

The cylinder has a circular floor area of:
pi*r^2
= pi*(70/2)^2
= 1225pi
We'll use this value later. Let x = 1225pi

The area of the curved walls, aka lateral area of the cylinder, is:
(circumference of circular base)*(height)
= (pi*diameter)*(height)
= (pi*70)*(84)
= 5880pi
We'll use this value later. Let y = 5880pi

Then the last piece to consider is the lateral surface area of the cone carved out of the cylinder.
We can think of this as an "inverted" surface area of sorts, since we're effectively looking at the inside wall of the cone we carved out.

The lateral surface area of a cone is
pi*r*L
where
L = slant height of the cone
L = sqrt(r^2+h^2) due to the pythagorean theorem

We can update that formula to get
pi*r*sqrt(r^2+h^2)
= pi*(70/2)*sqrt((70/2)^2+84^2)
= 3185pi
We'll use this value later. Let z = 3185pi

The last thing to do is add the results we got to determine the entire surface area of this strange 3D shape.
x+y+z = 1225pi+5880pi+3185pi
= (1225+5880+3185)pi
= 10290pi

The total exact surface area of this strange 3D shape is 10290pi square cm

I'll let the student compute the approximate version of this value.

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