Newton's law of cooling states that d x d t = − k ( x − a ) where x is the temperature, t is time, a is the ambient temperature, and k > 0 is a constant. suppose that a = a 0 cos ( ω t ) for some constants a 0 and ω . that is, the ambient temperature oscillates (for example night and day temperatures). a) find the general solution. b) in the long term, will the initial conditions make much of a difference? why or why not?

Askydiver weighing 140 lb (including equipment) falls vertically downward from an altitude of 19,000 ft and opens the parachute after 18 s of free fall. assume that the force of air resistance, which is directed opposite to the velocity, is of magnitude 0.55|v| when the parachute is closed and is of magnitude 14|v| when the parachute is open, where the velocity v is measured in ft/s. assume that acceleration due to gravity has magnitude 32 ft/s/s; remember that weight is the product of mass and gravitational acceleration. (a) find the speed of the skydiver when the parachute opens. (b) find the distance fallen before the parachute opens. (c) what is the limiting velocity vl after the parachute opens? (d) determine how long the sky diver is in the air after the parachute opens. (e) plot the graph of velocity versus time from the beginning of the fall until the skydiver reaches the ground.

Madison pushes with a 210-n horizontal force on a 24-kg crate of coffee resting on a warehouse floor. the force of friction on the crate is 74 n. what is the acceleration?

Suppose a wire with a current is pushed upward by a magnetic field. how would the wire move if the direction of the current reversed? explain your answer.