Question

A cylindrical barrel has a radius of 7.8 m and a height of 8.2 m. Doubling which dimension(s) will quadruple the volume of the barrel?

height
neither height nor radius
both height and radius
radius

Answer 1

Answer: Choice D) Radius

The height stays the same.

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Step-by-step explanation:

Let's compute the volume of this cylinder before any dimension is doubled.

V = pi*r^2*h

V = pi*(7.8)^2*(8.2)

V = 498.888pi

That value is exact in terms of pi.

Next, we have these three cases to consider

• A) Double the height only
• B) Double the radius only
• C) Double both the height and the radius

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Case A) Double the height only

The height was 8.2, but it now doubles to 16.4. The radius stays the same at 7.8

V = pi*r^2*h

V = pi*(7.8)^2*(16.4)

V = 997.776pi

Divide this new volume over the previous volume calculated earlier. The pi terms cancel.

(997.776pi)/(498.888pi) = 2

This shows that doubling the height will double the volume.

We rule out case A because we want to quadruple the volume.

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Case B) Double the radius only

The radius originally was 7.8 meters but now it doubles to 15.6 meters. The height stays the same at 8.2 m.

V = pi*r^2*h

V = pi*(15.6)^2*(8.2)

V = 1995.552pi

This value is exact.

Dividing this over the first volume calculated gets us...

(1995.552pi)/(498.888pi) = 4

This shows that the volume has been quadrupled. Case B works out and shows us that the answer is between answer choice C or answer choice D.

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Case C) Double the radius and the height.

The old radius and height are 7.8 m and 8.2 m respectively. Those values double to 15.6 m and 16.4 m.

They lead to this volume:

V = pi*r^2*h

V = pi*(15.6)^2*(16.4)

V = 3991.104pi

Divide that over the first volume

(3991.104pi)/(498.888pi) = 8

This larger cylinder is 8 times larger in volume compared to the original. We rule out case C.

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In summary, doubling the radius only while keeping the height the same will quadruple the volume of the cylinder.

This is why the final answer is choice D.

Answer 2

Doubling the radius of this cylindrical barrel would quadruple the volume.

Step-by-step explanation:

The volume of a cylinder of radius
`r` and height
`h` is:

`V = \pi \, r^(2)\, h`.

If radius
`r` stays the same, volume
`V` would be proportional to height
`h`. That is,
`V = (\pi\, r^(2))\, h`. Doubling height
`h\!` would only double
`V\!`.

If height
`h` stays the same, volume
`V` would be proportional to
`r^(2)`, the square of the radius. That is,
`V = (\pi\, h)\, r^(2)`. Doubling radius
`r\!` would quadruple
`r^(2)\!` and thus quadruple
`V\!` as required.