Question

Archeollogists have discovered the ledgendary Golden Igloo of the Pribiloff Islands, a structure consisting of half-spherical shell with an inner diameter of 1.25 m and an out diameter of 1.45 m. What is the value of this antiquity, given that gold currently sells for $1,263/ounce and has a density of 19.3 g/cm^3? (Note that 1 ounce =31.1034768 g)

Answer 1

Archeologists have discovered the legendary Golden Igloo of the Pribiloff Islands, a structure consisting of half-spherical shell with an inner diameter of $1.25\mathrm{m}$ and an out diameter of $1.45\mathrm{m}$. What is the value of this antiquity, given that gold currently sells for $1,263/ounce and has a density of $19.3\mathrm{g/cm^3}$? (Note that 1 ounce = 31.1034768 g)

Solution:

First, let's calculate volume of Igloo shell. It could be represented as half of the difference between volume of sphere with diameter equal to outer diameter of Igloo and volume of sphere with diameter equal to inner diameter of Igloo.

Formula of the volume for sphere is: $V = \frac{4}{3}\pi r^3$, where $r$ - radius of the sphere. Radius is the half of diameter, so we could transform formula into:

$$V = \frac {4}{3} \pi \left(\frac {d}{2}\right) ^ {3} = \frac {4}{24} \pi d ^ {3} = \frac {1}{6} \pi d ^ {3}.$$

So the shell volume is:

$$V _ {s} = \frac {1}{2} \left(\frac {1}{6} \pi d _ {o} ^ {3} - \frac {1}{6} \pi d _ {i} ^ {3}\right) = \frac {1}{2} * \frac {1}{6} \pi \left(d _ {o} ^ {3} - d _ {i} ^ {3}\right) = \frac {1}{12} \pi \left(d _ {o} ^ {3} - d _ {i} ^ {3}\right) = \frac {1}{12} * 3.141592 * ((145\ cm) ^ {3} - (125\ cm) ^ {3}) = 286801.1697\ cm ^ {3}.$$

(Note: $d_0$ and $d_i$ are outer and inner diameter of sphere respectively. We are converting meters to centimeters in equation to match density units.)

Now we can calculate value of this antiquity. For that we are finding mass of shell first:

$$m _ {s} = \rho * V _ {s} = 19.3\ \frac {g}{cm ^ {3}} * 286801.1697\ cm ^ {3} = 5535262.575\ g$$

Then, we are converting this mass from grams to ounces:

1 ounce = 31.1034768 g;

x ounce = 5535262.575 g;

$$x = \frac {5535262.575\ g * 1\ ounce}{31.1034768\ g} = 177962.8243\ ounces$$

Finally, we are calculating value of this antiquity:

Value = 177962.8243 ounces * 1263 $/ounce = 224767047.09$

Answer:

Value of the Golden Igloo is 224,767,047.09$ (224 million 767 thousand 47 dollars and 9 cents).

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