Introduction:
To calculate the maximum distance of a satellite from the center of the Earth, we need to consider the satellite's initial speed, escape speed, and the concept of circular motion.

Escape Speed:
The escape speed from Earth's surface is the minimum speed required for an object to escape the gravitational pull of the Earth. It can be calculated using the formula:

Escape speed = √(2 * G * M / R)

Where G is the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.

Initial Speed of the Satellite:
The satellite is projected with a speed of 5/6 times the escape speed from the Earth's surface. Let's denote this initial speed as V.

Circular Motion:
When a satellite is in orbit around the Earth, it follows a circular path due to the gravitational force acting as the centripetal force. The centripetal force can be calculated using the formula:

Centripetal force = (m * V^2) / R

Where m is the mass of the satellite, V is its speed, and R is the distance of the satellite from the center of the Earth.

Maximum Distance from the Center of the Earth:
To find the maximum distance, we need to consider the condition when the centripetal force is equal to the gravitational force between the satellite and the Earth. We can equate these two forces:

(m * V^2) / R = (G * m * M) / R^2

Simplifying the equation, we get:

V^2 = (G * M) / R

Now, we can substitute the value of V as (5/6) times the escape speed:

(5/6)^2 * (Escape speed)^2 = (G * M) / R

Simplifying further:

25/36 * (Escape speed)^2 = (G * M) / R

Dividing both sides by (Escape speed)^2:

25/36 = (G * M) / ((Escape speed)^2 * R)

Simplifying and rearranging the equation:

R = (36 * G * M) / (25 * (Escape speed)^2)

Therefore, the maximum distance of the satellite from the center of the Earth is given by the formula above.

Conclusion:
The maximum distance of the satellite from the center of the Earth can be calculated using the formula R = (36 * G * M) / (25 * (Escape speed)^2), where G is the gravitational constant, M is the mass of the Earth, and Escape speed is the speed required to escape Earth's gravitational pull.