Gravity is a force that attracts objects towards the center of the Earth. The acceleration due to gravity, denoted by g, is a constant value that represents the rate at which an object accelerates when falling freely under the influence of gravity. The value of g on the surface of the Earth is approximately 9.8 m/s².

When an object is at a certain altitude above the Earth's surface, the gravitational force acting on it decreases. This is because the distance between the object and the Earth's center increases, resulting in a weaker gravitational attraction. As a result, the acceleration due to gravity decreases.

To calculate the acceleration due to gravity at a given altitude, we can use the formula for gravitational acceleration:

g' = (GM) / (R + h)²

Where:
g' is the acceleration due to gravity at the given altitude
G is the gravitational constant
M is the mass of the Earth
R is the radius of the Earth
h is the altitude above the Earth's surface

In this case, the question asks us to find the altitude at which the acceleration due to gravity is half of its value on the surface of the Earth. Let's denote this altitude as h'.

We can set up the following equation:

(g / 2) = (GM) / (R + h')²

To simplify the equation, we can divide both sides by g and multiply both sides by (R + h')²:

1/2 = (R + h)² / R²

Taking the square root of both sides gives us:

√(1/2) = (R + h') / R

Simplifying further:

√(1/2) = (R / R) + (h' / R)

√(1/2) = 1 + (h' / R)

Subtracting 1 from both sides gives us:

√(1/2) - 1 = h' / R

Simplifying further:

h' / R = (√2 - 1)

Multiplying both sides by R gives us:

h' = R(√2 - 1)

Given that R is 4000 miles, we can substitute this value into the equation:

h' = 4000(√2 - 1) ≈ 1600 miles

Therefore, the correct answer is option 'C' - 1600 miles.