SOLUTION: A cone has a volume of 60 cubic cm. Determine the volume of a cylinder with the same circular base as the cone, but the height is 10 times taller.

Question 1203353: A cone has a volume of 60 cubic cm. Determine the volume of a cylinder with the same circular base as the cone, but the height is 10 times taller.
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: 1800 cubic cm

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Here's one approach
A second approach is talked about in the next section.

r = radius of the cone = radius of the cylinder
We know this because both 3D objects have the same circular base.

h = height of the cone
10h = height of the cylinder

volume of a cone = (1/3)*pi*(radius)^2*(height)
Vcone = (1/3)*pi*r^2*h

volume of a cylinder = pi*(radius)^2*(height)
Vcylinder = pi*r^2*(10h)

We see that "pi*r^2" is involved with both formulas. Let's replace that with B to represent the area of the base.

Vcone = (1/3)*pi*r^2*h
Vcone = (1/3)*Bh
and
Vcylinder = pi*r^2*(10h)
Vcylinder = B(10h)
Vcylinder = 10Bh

Both now involve Bh
Let's isolate Bh in the 1st formula
Vcone = (1/3)*Bh
Bh = 3*Vcone

Then we can use substitution
Vcylinder = 10Bh
Vcylinder = 10*(3*Vcone)
Vcylinder = 10*(3*60)
Vcylinder = 1800 cubic cm is the final answer

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Another approach

Let's consider making the cone 10 times taller.
We'll keep the circular base the same.

This will mean,
new cone volume = 10*(old cone volume)
new cone volume = 10*(60)
new cone volume = 600 cubic cm

Why does this work?
Well consider a rectangular prism of length, height, width of L,W,H
We'll scale H to 10H to make it 10 times taller. Keep everything else the same.
old volume = LWH
new volume = LW(10H) = 10LWH = 10*(old volume)
This idea can be applied to the cone because we can break the cone into very small cubic blocks.

Then recall the connection between a cylinder and a cone.

cone volume = (1/3)*(cylinder volume)
which rearranges to
cylinder volume = 3*(cone volume)

So we can fit 3 cones in a cylinder when these two conditions are true
The circular base areas are the same. This is equivalent to saying "The radii are the same" (or "the diameters are the same").
The height of the cone and cylinder are the same.
So you can probably see why I scaled the cone up so its height is 10 times taller.

From here we then write:
cylinder volume = 3*(new cone volume)
cylinder volume = 3*(600)
cylinder volume = 1800 cubic cm

Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.

When professional Math writers formulate Math problems in English,
they NEVER use the form " 10 times taller ", because such form is ambiguous in English.

The standard form to use in Math problems in English for such situations
is " 10 times as tall as . . . ".

        The strict formal meaning of the form " 10 times taller than . . . "
        in English is " 11 times as tall as . . . ".

        You will be very surprised, but it is so, to great distinction from other languages,
        as well as from your native language.

To learn more, read from this link

https://english.stackexchange.com/questions/221323/has-the-illogical-three-times-bigger-replaced-three-times-as-big-in-common-u

This error that you make in your post is a standard/typical mistake for newcomers:
they ALL do this mistake, until somebody will disclose them the truth.

////////////////

As for the solution, below it is presented in short form.

If "r" is the radius of the base of a cone and "h" is the cone's height, then the volume of a cone is = . From the problem, we are given that = = 60 cm^3. It implies that = 3*60 = 180 cm^3. This formula says that the volume of a cylinder with the same base and the same height is 180 cm^3. But since our cylinder is 10 times as tall as the cone, we should take 180 cm^3 ten times, which gives us the ANSWER. The volume of the cylinder under the problem's question is 1800 cm^3.

Solved.

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