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Step-by-Step Solution
Step 1: Identify Known Values
• Gravitational potential, $V = -5.4 \times 10^7\,\text{J}\,\text{kg}^{-1}$
• Acceleration due to gravity, $g = 6.0\,\text{m}\,\text{s}^{-2}$
• Radius of the Earth, $R = 6400\,\text{km}$
• Let $h$ be the height above the Earth's surface.
Step 2: Write the Relevant Equations
The gravitational potential at a distance $(R + h)$ from the center of the Earth is given by:
$$
V = -\frac{GM}{R + h}
$$
The acceleration due to gravity at the same distance is:
$$
g = \frac{GM}{(R + h)^2}
$$
Step 3: Relate the Two Equations by Division
Divide the expression for $V$ by the expression for $g$:
$$
\frac{V}{g}
= \frac{-\frac{GM}{R + h}}{\frac{GM}{(R + h)^2}}
= - (R + h).
$$
Substitute the given values for $V$ and $g$:
$$
-(R + h) = \frac{-5.4 \times 10^7}{6}.
$$
Step 4: Solve for $(R + h)$
$$
R + h = \frac{5.4 \times 10^7}{6}.
$$
Calculate the right-hand side:
$$
R + h = 9.0 \times 10^6 \,\text{m}.
$$
(noting that $5.4 \times 10^7 \div 6 = 9.0 \times 10^6$)
Since $1\,\text{km} = 1000\,\text{m}$, convert this distance to kilometers:
$$
R + h = 9000\,\text{km}.
$$
Step 5: Find the Height $h$ Above the Earth's Surface
We know $R = 6400\,\text{km}$, so:
$$
h = (R + h) - R = 9000 - 6400 = 2600\,\text{km}.
$$
Therefore, the height above the Earth's surface is $2600\,\text{km}$.
