Question

Suppose you have a pair of Seven League Boots. These are magic boots that enable you to stride along a distance of 7.0 leagues with each step. (a) If you march along at a military march pace of 120 paces/min, what will be your speed in \(\mathrm{km} / \mathrm{h}\) ? (b) Assuming you could march on top of the oceans when you step off the continents, how long (in minutes) will it take you to march around the Earth at the equator? (1 league $=3 \mathrm{mi}=4.8 \mathrm{km} .$ )

Short answer

Question: Using the Seven League Boots, which allow one to walk 7.0 leagues (4.8 km per league) with each step and maintaining a marching pace of 120 paces/min, calculate (a) the speed in km/h and (b) the time taken to march around the Earth at the equator, assuming you can step on the oceans. Answer: (a) The speed when using the Seven League Boots is 241,920 km/h, and (b) marching around the Earth at the equator will take approximately 9.9 minutes.

Step-by-step solution

Convert leagues to kilometers

We are given the conversion factors: 1 league = 3 mi = 4.8 km. So, 1 step with the Seven League Boots covers 7.0 leagues, which is equivalent to: $$7.0 \; \text{leagues} \times \frac{4.8 \; \text{km}}{1 \; \text{league}} = 33.6 \; \text{km}$$

Calculate the distance covered per minute (km/min)

We know that 120 steps are taken per minute, so we can find the distance covered in one minute: $$120 \; \text{steps/min} \times \frac{33.6 \; \text{km}}{1 \; \text{step}} = 4032 \; \text{km/min}$$

Convert the speed to km/h

To convert km/min to km/h, we need to multiply km/min by the conversion factor of 60 min/h: $$4032 \; \text{km/min} \times \frac{60 \; \text{min}}{1 \; \text{h}} = 241,920\; \text{km/h}$$

Find the circumference of Earth at the equator

The average radius of the Earth is about 6371 km. To find the circumference at the equator, we can use the formula \(C = 2\pi r\): $$C = 2 \pi \times 6371 \; \text{km} \approx 40,075 \; \text{km}$$

Calculate the time taken to march around the Earth

To find the time taken to march around the Earth, we need to divide the total distance (Earth's circumference) by the speed in km/h: $$\text{Time} = \frac{40,075 \; \text{km}}{241,920 \; \text{km/h}} \approx 0.165 \; \text{h}$$

Convert the time to minutes

To convert the time to minutes, we need to multiply the time in hours by a conversion factor of 60 min/h: $$0.165 \; \text{h} \times \frac{60 \; \text{min}}{1 \; \text{h}} \approx 9.9 \; \text{min}$$ The final results are (a) the speed when using the Seven League Boots will be 241,920 km/h, and (b) marching around the Earth will take approximately 9.9 minutes.