# Chocolate Math Challenge

Our T. rex in the Blue Wing has been watching visitors come in and out of the Chocolate exhibit for weeks now. He’s getting hungry for chocolate!

### How many chocolate bars can T. rex eat?

Let’s assume T. rex is really, really hungry, and can fill his entire body with chocolate bars. It may help to consider the volume of T. rex and the volume of a bar of chocolate. Below is some useful information to add to your mathematical toolkit!

## Measurements

A chocolate bar is about 6 inches long, 2 inches wide, and a half an inch tall.

The Blue Wing's T. rex is 44 feet long (nose to tip of tail) and 15 feet tall (feet to top of head)*. Let's assume his width is a third of his height: 5 feet.

* Fun fact: In this case, we know exact height. However, scientists often measure two legged dinosaurs at the hip, because we don’t know how much they actually "stood up."

## Equations

1 foot = 12 inches

Volume of a rectangular prism = (Height)(Width)(Depth)

## Need a hint?

Puzzle solution steps (in three parts) were posted the following dates:

April 4, 2017: First solution step posted

April 11, 2017: Second solution step posted

April 25, 2017: Third and final solution step posted

-   Solution Step 1

### 1

Determine the volume of a chocolate bar.

A chocolate bar resembles a shape we know the volume of — a rectangular prism! The volume is calculated by multiplying height, width, and depth. Let’s estimate a chocolate bar is 5 inches long, 2 inches wide, and a half inch tall. Multiply those numbers to find the volume of a chocolate bar:

$$V_1 = hwd$$

$$V_1 = (0.5 in)(2 in)(6 in)$$

$$V_1 = 6 in^3$$

-   Solution Step 2

### 2

Determine the volume of a T. rex.

There is no formula for the size of a T. rex. There isn't even a simple shape that resembles a dinosaur, so what now? Well, I'm sure you've seen an animal in a cage, which is a rectangular prism like the chocolate bar. Imagine a T. rex in a huge glass exhibit case. About what percentage of that case would a T. rex fill? Just to get things moving, let's say 40% (or, two fifths). We know from the puzzle prompt that T. rex was 15 feet tall, 5 feet wide, and 40 feet long.

$$V_2 = hwd \left(\frac{2}{5}\right)$$

$$V_2 = (15 ft)(5 ft)(44 ft) \left(\frac{2}{5}\right)$$

$$V_2 = 1320 ft^3$$

-   Solution Step 3

### 3

Determine the number of chocolate bars that fit in a T. rex.

You can find the number of chocolate bars that fit into a T. rex by dividing the volume of the dino by the volume of a single chocolate bar. But don't forget the unit conversion! In the last two steps, we measured the chocolate bar in inches and the T. rex in feet. There are 12 inches to a foot, but because we're talking volume we need to remember to cube 12. So, the total number of chocolate bars equals:

$$Bars = \left(\frac{V_2}{V_1}\right) \left(\frac{12 in}{ 1 ft}\right)^{3}$$

$$Bars = \left(\frac{1320 ft^3}{6 in^3}\right) \left(\frac{1728 in^3}{1 ft^3}\right)$$

Bars = 380,160

### How did our audience think about it?

Since this is a "back of the envelope" or "Fermi" question, there is no correct answer. The question itself is a little silly and unrealistic, but that's the fun, right? Math challenge submissions were reviewed using the following key ideas: 1) seeing chocolate as a geometric shape 2) approximating the volume of a dinosaur using a shape(s) we already know the volume of and 3) converting between feet and inches.

Congratulations to the 19 people who calculated an answer between 100,000 and 1 million! (This was where the foot-to-inches conversation was key.):

Philip, Parker, Sergey, Phil, Matthew, Nathan, Addison, Isla, Niraagi, Isla, Sarah, Lily, Erdem, Robel, Richie, Isaiah, Steve, Margy, Steven

And a special mention for Courtney, who explained in her answer that you can describe T. rex as filling some percentage of a cube. We estimated T. rex's shape fills 40% and Courtney estimated 30%.

We’d like to recognize those who considered alternative ways to look at the math challenge, or who questioned it entirely. Thanks for sharing your creative ideas!

"T. rex is an irregular shape and doesn't take up all of the rectangular prism. To get a more accurate calculation of how many chocolate bars fit inside the T. rex, you could find its volume using water displacement." — Kim

"I modeled the body of the T. rex to be a cone plus a cylinder, each of diameter 5 feet and length 44 feet, and its legs to be cylinders of length 5 feet and diameter 1 foot." — Phil

"If I break the T. rex shape into pieces, the tail is approximately a rectangular pyramid about 18 feet from base to tip attaching to the body with a base of about 3 feet x 3 feet… If I add the volume of the legs to the volume of the head and neck, I can get a sort of cylinder averaging a little less than 5 feet in diameter and 26 feet long…Note: The stomach of T. rex would be MUCH smaller." — Margy

"0! T. rex do not eat chocolate. April Fool's?" — Elena

"My dad is telling me that this is a trick question because the T. rex can only fit so many candy bars in his belly at a time without getting very sick and eating his volume in candy would actually kill him if all done at once. Who is right here?" — Madisynn

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