At the time this problem was written, the price of gold was about $433 per ounce

Question

At the time this problem was written, the price of gold was about $433 per ounce, while that of platinum was about $885 an ounce. The "ounce" in this case is the troy ounce, which is equal to 31.1035 g . (The more familiar avoirdupois ounceis equal to 28.35 g .) The density of gold is 19.3 g/cm3 and that of platinum is 21.4 g/cm3 .

Part A

If you find a spherical gold nugget worth 1.00 million dollars, what would be its diameter?

Part B

How much would a platinum nugget of this size be worth?

At the time this problem was written, the price of gold was about $433 per ounce, while that of platinum was about $885 an ounce. The "ounce" in this case is the troy ounce, which is equal to 31.1035 g . (The more familiar avoirdupois ounceis equal to 28.35 g .) The density of gold is 19.3 g/cm3 and that of platinum is 21.4 g/cm3 .

Part A

If you find a spherical gold nugget worth 1.00 million dollars, what would be its diameter?

Part B

How much would a platinum nugget of this size be worth?

Explanation / Answer

PART A:
Let's first figure out how many ounces of gold we would need.
$1,000,000 / $433/ounce = 2309.47 troy ounces

Next, convert that to grams:
2309.47oz. * 31.1035g/oz. = 71832.6g

Next, let's figure out the volume:
71832.6g / 19.3g/cm^3 = 3721.9 cm^3

Now use the formula for the volume of a sphere to figure out the radius:
V = 4/3 r^3
3721.9 = 4/3 r^3
r^3 = 3721.9/ (4/3) = 888.54
r = (888.54)^(1/3) <-- cube root
r = 9.4 cm

The diameter is twice the radius:
d = 2r
d = 2*9.4cm
d = 18.8 cm

PART B:
We can now work the problem in the opposite direction to figure out the value of a sphere of platinum. The volume is the same:
V = 3721.9 cm^3

The density is 21.4g/cm^3:
3721.9* 21.4g/cm^3 = 79648.66 g

Convert this back to troy ounces:
79648.66g / 31.1035g/oz = 2560.76 ounces

And now figure the value in dollars:
2560.76 oz * $885/oz = 2266272.6

Answer: $ 2266272.6

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