Question

The average depth of the oceans is about \(4 \mathrm{km}\) and oceans cover about \(70 \%\) of Earth's surface. Make an order-of-magnitude estimate of the volume of water in the oceans. Do not look up any data in books. (Use your ingenuity to estimate the radius or circumference of Earth.)

Short answer

Question: Estimate the volume of water in the Earth's oceans, given that they cover 70% of the Earth's surface and have an average depth of 4 km. Do not look up any data; you may estimate the Earth's circumference to be about 40,000 km. Answer: The volume of water in the Earth's oceans is approximately \(1.4 \times 10^9\: \mathrm{km}^3\).

Step-by-step solution

Calculate the ocean surface area

To find the surface area of the oceans, we need to multiply the Earth's total surface area by the percentage that oceans cover. The Earth's total surface area can be calculated using the formula for the surface area of a sphere: \(4\pi r^2\), where r is the Earth's radius. We are not allowed to look up Earth's radius, but we can estimate it. We know that the Earth's circumference is close to \(40,000 \mathrm{km}\), from which we can estimate the radius using the formula for the circumference of a circle, \(C = 2\pi r\). Estimating \(r\) we have: \(40,000 = 2\pi r\) \(r = \frac{40,000}{2\pi} \approx 6\, 370 \mathrm{km}\) Now, we can calculate the Earth's total surface area: \(A_\text{Earth} = 4\pi r^2 \approx 4\pi (6370)^2 \approx 510\, 000\, 000\: \mathrm{km}^2\) Next, calculate the ocean surface area: \(A_\text{ocean} = 0.7 \times A_\text{Earth} \approx 0.7 \times 510\,000\,000 \approx 357\, 000\, 000\: \mathrm{km}^2\)

Calculate the volume of the oceans

To estimate the volume of water in the oceans, we can multiply the ocean surface area by the average depth of the oceans: \(V_\text{ocean} = A_\text{ocean} \times d^*\) where \(d^* = 4 \mathrm{km}\) is the given average depth of the oceans. Calculating the volume: \(V_\text{ocean} \approx 357\, 000\, 000 \times 4 \approx 1\, 428\, 000\, 000\: \mathrm{km}^3\) So, an order-of-magnitude estimate of the volume of water in the oceans is \(1.4 \times 10^9\: \mathrm{km}^3\).