To solve this problem, we need to consider the following:

1. Escape velocity: The escape velocity at Earth’s surface is approximately 11.2 km/s. So, the initial velocity of the body (3/4)th of escape velocity will be (3/4)*11.2 km/s = 8.4 km/s.

2. Potential energy: As the body moves upward, its kinetic energy converts into potential energy. At the maximum height, all the kinetic energy will be converted into potential energy.

3. Gravitational potential energy: The potential energy due to gravity at a distance R from the Earth’s center is given by: Potential Energy = -GMm/R, where G is the gravitational constant, M is the Earth’s mass, and m is the mass of the body.

4. Distance: We need to find the maximum height (R) reached by the body.

Solution:

1. At the maximum height, the kinetic energy becomes zero, and the total energy is converted into potential energy. Therefore, we can equate the initial kinetic energy and the final potential energy:

1/2 mv^2 = GMm/R

where v is the initial velocity, m is the mass of the body, G is the gravitational constant, and M is the Earth’s mass.

1. Substitute the values of v, G, M, and simplify:

1/2 (m)(8.4 km/s)^2 = 6.6743 x 10^-11 Nm^2/kg^2 (5.972 x 10^24 kg) (m)/R

1. Cancel out common factors and arrange:

R = (m)(8.4 km/s)^2 / (2 x 6.6743 x 10^-11 Nm^2/kg^2 x 5.972 x 10^24 kg)

1. This equation gives the height in terms of the body’s mass (m). However, the problem doesn’t specify the mass of the body. Therefore, we cannot calculate the exact height without knowing the body’s mass.

Conclusion:

The body will reach a certain height based on its mass. However, we need to know the body’s mass to calculate the specific height it reaches from the Earth’s center.