1 Colonizing a Black Hole (10 points)

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The year is 2237. Having developed interstellar travel and begun vast mining

operations, humanity has decided to begin a scientific operation to build a base

around the small black hole M231 X-3. The goal is to build a hollow sphere of

metal around the black hole, to act as an artificial planet from which research

into the black hole's nature can be performed. But in order to do so, scientists

and engineers need to perform some crucial calculations in order to determine

the scale of the project.

We have sent a space probe to measure and photograph M231 X-3. The

probe was, ultimately, destroyed, but not before sending us valuable data on

the black hole's characteristics. We have created a standardized rectangular

coordinate grid with which to map out M231 X-3 and surrounding structures,

with the black hole itself centered at the origin.

(a) (2 points) According to the probe's measurements, the following equation

dictates the gravitational field created by M231 X-3 at a given position in

space:

300000

9 = 2² + y² + 2²

(Here the gravitational field is given in m/s², and the positional coordinates

are given in km.)

List the independent and dependent variable(s) in this equation.