Answer the following questions:

1. A car is being compacted into a rectangular solid. The volume is decreasing at a rate of 22 m³/sec. The length and width of the compactor are square, but the height is not the same length as the length and width. If the length and width walls move toward each other at a rate of 0.250.25 m/sec, find the rate at which the height is changing when the length and width are 22 m and the height is 1.51.5 m.

2. An airline sells tickets from Tokyo to Detroit for $1200.$1200. There are 500500 seats available and a typical flight books 350350 seats. For every $10$10 decrease in price, the airline observes an additional five seats sold. What should the fare be to maximize profit? How many passengers would be onboard?

3. A rocket is launched into space; its kinetic energy is given by 𝐾(𝑡)=(12)𝑚(𝑡)𝑣(𝑡)2,K(t)=(12)m(t)v(t)2, where 𝐾K is the kinetic energy in joules, 𝑚m is the mass of the rocket in kilograms, and 𝑣v is the velocity of the rocket in meters/second. Assume the velocity is increasing at a rate of 1515 m/sec² and the mass is decreasing at a rate of 1010 kg/sec because the fuel is being burned. At what rate is the rocket’s kinetic energy changing when the mass is 20002000 kg and the velocity is 50005000 m/sec? Give your answer in mega-Joules per second (MJ/s), which is equivalent to 106106 J/s.