Final answer:
The height the elevator reaches, calculated using work and energy principles and given the tension in the cable and the final speed, is 17.3 meters.
Step-by-step explanation:
To solve for the height the elevator reaches using work and energy, we start by finding the kinetic energy (KE) the elevator has at its final speed. The formula for kinetic energy is KE = (1/2)mv2, where m is mass and v is velocity. Plugging in the values, KE = (1/2)(856 kg)(18.4 m/s)2 = 144837.12 J.
The work done by the tension in the cable is equal to the change in kinetic energy plus the work done against gravity (since the elevator also moves upward against the force of gravity). The work done against gravity is the gravitational potential energy (GPE), which is GPE = mgh, where g is the acceleration due to gravity (9.81 m/s2) and h is the height. We can set up the equation: Work = KE + GPE = (1.47×104 N)h.
Equating the KE to the work done against gravity and the cable's tension gives us: 144837.12 J = (1.47×104 N)h – (856 kg)(9.81 m/s2)h. Solving for h we get: h = 144837.12 J / ((1.47×104 N) - (856 kg)(9.81 m/s2)) = 17.306 meters, which can be approximated to 17.3 m.