Question

It has been estimated that there is \(4 \times 10^{-4} \mathrm{mg}\) of gold per liter of seawater, At a price of \(\$ 19.40 / \mathrm{g}\), what would be the value of the gold in \(1 \mathrm{~km}^{3}\left(1 \times 10^{15} \mathrm{~cm}^{3}\right)\) of the ocean?

Short answer

The value of the gold in 1 km³ of the ocean is \(7.76 \times 10^6\) dollars.

Step-by-step solution

- Determine gold concentration in mg in a cubic kilometer

Given the concentration of gold is \(4 \times 10^{-4}\) mg per liter. First, convert the volume of the sea water from cubic kilometers to liters. Since \(1 \text{ km}^3 = 1 \times 10^{15} \text{ cm}^3\) and \(1 \text{ cm}^3 = 1 \text{ ml}\), \(1 \text{ km}^3 = 1 \times 10^{15} \text{ ml}\). Because \(1000 \text{ ml} = 1\) liter, \(1 \text{ km}^3 = 1 \times 10^{12}\) liters.

- Calculate the total gold in kg

To find the total mass of gold, multiply the volume by the concentration:\(1 \times 10^{12}\) liters \(\times 4 \times 10^{-4}\) mg/liter = \(4 \times 10^{8}\) mg. Since \(1,000,000\) mg = 1 kg, \(4 \times 10^{8}\) mg = \(4 \times 10^2\) kg.

- Convert gold mass to grams

Knowing that each kilogram has \(1,000\) grams, \(4 \times 10^2\) kg = \(4 \times 10^5\) grams.

- Calculate the value of the gold

Finally, at a price of \(19.40\) per gram, multiply the total grams by the price to find the value: \(4 \times 10^5\) \(\times 19.40 = 7.76 \times 10^6\) dollars.

Key concepts

Gold concentration calculation

To calculate the gold concentration in a given volume of seawater, you need to know the concentration in smaller units first. In our example, the given concentration is 4 x 10^-4 mg of gold per liter. This means for every liter (1 liter = 1000 ml = 1000 cm³) of seawater, you have 0.0004 mg of gold. Knowing these smaller units helps us calculate larger volumes accurately.

Unit conversion

When dealing with large volumes like a cubic kilometer (km³), it's important to convert units step-by-step. In the problem, 1 km³ is equivalent to 1 x 10^15 cm³. Since 1 cm³ equals 1 ml and 1000 ml equals 1 liter, we convert 1 km³ to liters: 1 x 10^15 cm³ = 1 x 10^15 ml = 1 x 10^12 liters. Accurate unit conversion ensures that you are working with comparable quantities.

Mass-to-value conversion

After determining the total gold mass, it's essential to convert this mass into a financial value. First, calculate the total gold in mg and convert it to kg, then to grams, and finally to dollars. In this problem, 4 x 10^8 mg is converted to 400 kg (since 1,000,000 mg = 1 kg) and then to 400,000 grams (since 1 kg = 1000 grams). The financial value is found by multiplying the grams by the price per gram: 400,000 grams x \(19.40 = \)7.76 million.

Steps in problem solving

Breaking down a complex problem into manageable steps is key for accurate results. Here are the main steps:
1. **Identify given quantities and required conversions:** For our example, we start with the gold concentration and needed volume conversions.
2. **Convert volumes and units accurately:** Convert the cubic kilometer to liters, ensuring all units match.
3. **Calculate the total mass:** Multiply the volume by concentration to find the total mass of gold in mg, then convert to kg and grams.
4. **Determine final value:** Finally, convert the total gold mass into monetary value using the given price per gram. By following these steps, you simplify complex calculations and reduce errors.