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We can find the equation that relates the period (T), radius (r), and orbital speed (v) of the satellite with the following formula: \[v = \frac{2 \pi r}{T}\] This equation comes from the fact that the circumference of the orbit is \(2 \pi r\), and the satellite travels this distance in a time period T.

Next, we need to apply Newton's second law. The gravitational force acting on the satellite is the centripetal force that maintains its circular orbit: \[F_{gravity} = F_{centripetal}\] The gravitational force can be represented as: \[F_{gravity} = G \frac{M_{earth}m_{satellite}}{r^2}\] where \(G\) is the gravitational constant, \(M_{earth}\) is the mass of the Earth, and \(m_{satellite}\) is the mass of the satellite. The centripetal force can be represented as: \[F_{centripetal} = m_{satellite}\frac{v^2}{r}\] Now, we can equate the two forces: \[G \frac{M_{earth}m_{satellite}}{r^2} = m_{satellite}\frac{v^2}{r}\]

We need to eliminate the speed (v) from the equations. First, solve the equation from Step 1 for v: \[v = \frac{2 \pi r}{T}\] Now, square the equation: \[v^2 = \frac{4 \pi^2 r^2}{T^2}\] Insert this expression for \(v^2\) into the equation from Step 2: \[G \frac{M_{earth}m_{satellite}}{r^2} = m_{satellite}\frac{4 \pi^2 r^2}{T^2 r}\] We can cancel the \(m_{satellite}\) from both sides: \[G \frac{M_{earth}}{r^2} = \frac{4 \pi^2 r^2}{T^2 r}\]

Now, solve the resulting equation for the radius (r): \[r^3 = \frac{G M_{earth} T^2}{4 \pi^2}\] Taking the cube root of both sides: \[r = \sqrt[3]{\frac{G M_{earth} T^2}{4 \pi^2}}\] Since the satellite is geosynchronous, its period (T) is equal to the Earth's rotational period, 24 hours, or 86400 seconds. Plug in the known values: \[r = \sqrt[3]{\frac{(6.674 \times 10^{-11} \text{ m}^3\text{ kg}^{-1}\text{ s}^{-2})(5.972 \times 10^{24} \text{ kg})(86400\text{ s})^2}{4 \pi^2}}\] Finally, calculate the value of r: \[r \approx 42,164 \text{ km}\] This is the orbital radius of a geosynchronous satellite. Note that this value includes the radius of the Earth; to find the altitude above the Earth's surface, subtract the Earth's radius: \[Altitude \approx 42,164 \text{ km} - 6,371 \text{ km} = 35,793 \text{ km}\]