D 0 Now p 338 16 Find the

D 0 Now: p. 338, #16 Find the exponential function through the two points. Initial value: Equation: Other point: Function: whose graph passes

Section 6. 4 b Applications of Exponential Change

Newton’s Law of Cooling Suppose that you just took a delicious hot pocket out of the microwave. The tasty treat will gradually cool to the temperature of the surrounding air… As it turns out, the rate at which the hot pocket’s temperature is changing at any given time is proportional to the difference between its temperature and the temperature of the surrounding medium!!! This leads us to derive NEWTON’S LAW OF COOLING (which, incidentally, works for warming as well)

Let T be the temperature of the object in question at time t, and Ts be the surrounding temperature. Then: Since , we can rewrite the equation: By the law of exponential change, the solution is

Newton’s Law of Cooling where T 0 is the temperature at time t = 0.

Newton’s Law of Cooling A hard-boiled egg at 98 C is put in a pan under running 18 C water to cool. After 5 minutes, the egg’s temperature is found to be 38 C. How much longer will it take the egg to reach 20 ? Define Variables: Law of Cooling: Substitute:

Newton’s Law of Cooling A hard-boiled egg at 98 C is put in a pan under running 18 C water to cool. After 5 minutes, the egg’s temperature is found to be 38 C. How much longer will it take the egg to reach 20 ? To find k, use the point (5, 38): The Final Equation:

Newton’s Law of Cooling A hard-boiled egg at 98 C is put in a pan under running 18 C water to cool. After 5 minutes, the egg’s temperature is found to be 38 C. How much longer will it take the egg to reach 20 ? Solve analytically: After about 13. 305 minutes, the egg will reach 20 degrees C.

Resistance Proportional to Velocity In many situations, the resistance encountered by a moving object (i. e. , from friction) is proportional to the object’s velocity… That is, the slower the object moves, the less its forward progress is resisted by the air through which it passes… To model such a situation, we’ll start with Newton’s Second Law of Motion…

Resistance Proportional to Velocity The resisting force opposing the motion: Force = mass x acceleration If this force is proportional to the velocity, then: or This is a differential equation of exponential change… The solution:

Guided Practice For a 50 -kg ice skater, the k in the previous equation is about 2. 5 kg/sec. How long will it take the skater to coast from 7 m/sec to 1 m/sec? How far will the skater coast before coming to a complete stop? First, the general model:

Guided Practice For a 50 -kg ice skater, the k in the previous equation is about 2. 5 kg/sec. How long will it take the skater to coast from 7 m/sec to 1 m/sec? How far will the skater coast before coming to a complete stop? Now, we want the value of t when v = 1. The skater will reach 1 m/sec from 7 m/sec after about 38. 918 sec of coasting

Guided Practice For a 50 -kg ice skater, the k in the previous equation is about 2. 5 kg/sec. How long will it take the skater to coast from 7 m/sec to 1 m/sec? How far will the skater coast before coming to a complete stop? To find distance, we need the integral of velocity:

Guided Practice For a 50 -kg ice skater, the k in the previous equation is about 2. 5 kg/sec. How long will it take the skater to coast from 7 m/sec to 1 m/sec? How far will the skater coast before coming to a complete stop? Assuming s = 0 when t = 0, we have

Guided Practice For a 50 -kg ice skater, the k in the previous equation is about 2. 5 kg/sec. How long will it take the skater to coast from 7 m/sec to 1 m/sec? How far will the skater coast before coming to a complete stop? Finally, for distance: Find Mathematically, s never quite reaches 140. But for all practical purposes, the skater comes to a complete stop after traveling 140 m…

A General Pattern The distance traveled by a moving object that encounters resistance proportional to its velocity: The total distance traveled by this object:

Guided Practice Suppose a battleship has mass around 51, 000 metric tons (51, 000 kg) and a k value of about 59, 000 kg/sec. Assume the ship loses power when it is moving at a speed of 9 m/sec. (a) About how long will it take the ship’s speed to drop to 1 m/sec? The ship will reach 1 m/sec in about 1899. 296 seconds, or in about 31. 655 minutes

Guided Practice Suppose a battleship has mass around 51, 000 metric tons (51, 000 kg) and a k value of about 59, 000 kg/sec. Assume the ship loses power when it is moving at a speed of 9 m/sec. (b) About how far will the ship coast before it is dead in the water? The ship will coast for a distance of about 7779. 661 meters, or 7. 780 kilometers