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:

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.

Answer 2

Answer:  Choice D) Radius

The height stays the same.

======================================================

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

-------------------

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.

-------------------

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.

-------------------

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.

-------------------

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.