Find the work done in raising a mass of 50,000 kg from the surface of the Moon to a height of 200 km. (Check in Zill in the Applications Topic of the integral, mechanical work, work done against gravity)

To find the work done in raising a mass of 50,000 kg from the surface of the Moon to a height of 200 km, we can use the formula for work done against gravity:

W = \int_{r_1}^{r_2} F \cdot dr

Where:
- W is the work done,
- F is the force (weight of the mass) exerted vertically upward, which is F = mg ,
- r1 is the initial position at the surface of the Moon,
- r2 is the final position at a height of 200 km.

The force applied is equal to the weight of the mass:
F = mg

Where:
- m = 50,000 kg (mass of the object),
- g = 1.62 m/s^2 (acceleration due to gravity on the Moon).

We need to convert the distance to meters:
200 \, km = 200,000 \, m

Plugging in the values, we get:
W = \int_{0}^{200000} F \cdot dr = \int_{0}^{200000} mg \cdot dr

W = \int_{0}^{200000} 50000 \cdot 1.62 \cdot dr = 50000 \cdot 1.62 \cdot \int_{0}^{200000} dr

W = 50000 \cdot 1.62 \cdot (200000 - 0) = 50000 \cdot 1.62 \cdot 200000

W=50000\cdot1.62\cdot200000=1.62\times10^8\,J

Therefore, the work done in raising a mass of 50,000 kg from the surface of the Moon to a height of 200 km is 1.62\times10^8\,J .

\boxed{1.62\times10^8\,J}