Amounting to a Hill of Beans

The definition of the hill is argued, but is generally considered an increase of elevation no more than 1000 feet. The angle of repose for soybeans is 27 degrees. The price of a bushel of soybeans (as in, from a farmer) is around $14. With this, we can figure out the price of a hill of soybeans!

The formula for the volume of a cone is

$V = \frac{1}{3} \pi r^2 h$

so we just need to calculate the radius based on the angle of repose. We can do this by imagining an ABC right triangle where A is 1000 feet and angle BC is 27 degrees. With a simple trigonometry rule, we know

$\frac{1000}{\tan(27^\circ)} = B$

and therefore B = 1963 feet.

Now we can solve for the volume of the cone,

$V = \frac{1}{3} \pi (1963^2) (1000) = 4.04 \times 10^9 \text{ cubic feet}$

1 bushel is defined as 1.24 cubic feet, so the maximum volume of a hill of soybeans is 3.26 billion bushels.

At 14 dollars, that’s worth $45.6 billion dollars. Only 25 people today can say that they have amounted to more than a hill of beans. I guess your mean elementary school teacher was right: you’re never going to amount to a hill of beans.

Could we set the bar lower?

Amounting to more than a hill of beans is pretty tough! But we can say that “amounting to a hill of beans” is instead based on the minimum hill size. Let’s calculate that!

The smallest definition of a hill that I’ve found online is 100 feet of prominence, i.e., elevation above surroundings. Applying the same math for 100 feet instead of 1000 gets us:

$\frac{100}{\tan(27^\circ)} = 196$