How much hot chocolate powder would be necessary to make a bathtub full of hot chocolate?

11 January 2009

It's currently snowing outside, and as we all know, no drink is better suited for cold weather than hot chocolate. Sitting by the fireplace with a nice mug of...

Wait a minute, mug? That won't do. If we're going to have hot chocolate, we're going to go all out, and fill the largest container for liquid I have nearby: the bathtub.

I would suggest cleaning the tub a little more before acutally attempting to make hot chocolate in it.

How much hot chocolate powder would it take to do this?

First, we need to know how much hot chocolate we're making. We could get a rough estimate for the volume of the tub by measuring the width, length, and fill depth, but the tub is curved, so it wouldn't be very accurate. Rather than doing the calculations necessary for a curved container, we'll use another method to calculate the volume of the tub. We take a container of known volume (I found a 13-quart bin) and time how long it takes to fill it.

Then, we time how long it takes to fill the tub, and from there, if we assume the flow rate from the tub faucet doesn't change, we can easily calculate the tub's volume.

It took 62.4 seconds to fill the 13-quart bucket. I then filled the bathtub until just before the bottom of the overflow drain, which took 750.0 seconds.

This is how much hot chocolate we intend to make.

From these numbers, it's relatively simple to calculate the volume of water in the tub:

((13 quart )/(62.4 s ))=(v_t/(750.0 s))

v_t=156 quart

156 quart ((0.946 L )/(1 quart ))=148 L

So we want to make 148 L of hot chocolate. But how much powder will this take?

The recipe on the side of the hot chocolate powder suggests 2 heaping tablespoons per 175 mL. However, the nutrition facts say that the serving size is 2 tablespoons, so I'll go with that value, as "heaping" is a bit harder to pin down.

Now, adding chocolate to the water will increase the volume, and we don't want to overflow the tub. Countering that, heating up the water may cause some evaporation, which may reduce the volume. We want to make 148 L of hot chocolate, so how much water should we start with, taking the above into consideration?

Following the recommended ratio of 2 tablespoons of chocolate powder to 175 mL of water, I added 4 tablespoons of hot chocolate powder to 350 mL of water and microwaved it for 1 minute. The end result was 380 mL of hot chocolate.

So to make 148 L of hot chocolate:

((0.35 L water )/(0.38 L hot chocolate ))=(v_w/(148 L hot chocolate ))

v_w=136 L water

We would need to start with 136 L of water.

Now, how much hot chocolate powder do we need to mix in? Again, if we use 2 tablespoons of powder per 175 mL of water, then:

136 L ((2 tbsp )/(0.175 L ))=1554 tbsp

It would take 1554 tablespoons of hot chocolate powder to make a bathtub full of hot chocolate.

How much would that weigh?

The hot chocolate container contains 500 g of powder, and makes 18 two-tablespoon servings. So 36 tablespoons of hot chocolate powder weighs 500 g.

1554 tbsp ((500 g )/(36 tbsp ))=21583 g=21.6 kg

21.6 kg ((2.20 lb )/(1 kg ))=47.5 lb

So it would take 21.6 kg (47.5 lb) of hot chocolate powder to make a bathtub's worth of hot chocolate!

Note: everything here is an approximation, so significant figures will not receive an undue amount of focus. If this bothers you, round the numbers yourself.

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