Find the exponential function whose graph passes through the two points Initial value Equation Other point Function Applications of Exponential Change Section 64b Newtons Law of Cooling ID: 464885
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Slide1
D0 Now: p.338, #16
Find the exponential function whose graph passesthrough the two points.
Initial value:
Equation:
Other point:
Function:Slide2
Applications of Exponential Change
Section 6.4bSlide3
Newton’s Law of Cooling
Suppose that you just took a delicious hot pocket outof the microwave. The tasty treat will gradually coolto the temperature of the surrounding air…As it turns out, the rate at which the hot pocket’stemperature 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)Slide4
Let T be the temperature of the object in question at time
t,and T be the surrounding temperature. Then:
s
Since
,
we can rewrite the equation:
By the law of exponential change, the solution isSlide5
Newton’s Law of Cooling
where T is the temperature at time
t
= 0.
0Slide6
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 tobe
38 C. How much longer will it take the egg to reach 20 ?
Define Variables:
Law of Cooling:
Substitute:Slide7
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 tobe
38 C. How much longer will it take the egg to reach 20 ?
To find
k
, use the point (5, 38):
The Final Equation:Slide8
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 tobe
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.Slide9
Resistance Proportional to Velocity
In many situations, the resistance encountered by amoving object (i.e., from friction) is proportional to theobject’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’sSecond Law of Motion…Slide10
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:Slide11
Guided Practice
First, the general model:
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/secto 1 m/sec? How far will the skater coast before coming to acomplete stop?Slide12
Guided Practice
For a 50-kg ice skater, the
k
in the previous equation is about2.5 kg/sec. How long will it take the skater to coast from 7 m/secto 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 coastingSlide13
Guided Practice
For a 50-kg ice skater, the
k
in the previous equation is about2.5 kg/sec. How long will it take the skater to coast from 7 m/secto 1 m/sec? How far will the skater coast before coming to a
complete stop?To find distance, we need the integral of velocity:Slide14
Guided Practice
For a 50-kg ice skater, the
k
in the previous equation is about2.5 kg/sec. How long will it take the skater to coast from 7 m/secto 1 m/sec? How far will the skater coast before coming to a
complete stop?Assuming s
= 0 when t = 0, we haveSlide15
Guided Practice
For a 50-kg ice skater, the k in the previous equation is about2.5
kg/sec. How long will it take the skater to coast from 7 m/secto 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…Slide16
A General Pattern
The distance traveled by a moving object thatencounters resistance proportional to its velocity:
The total distance traveled by this object:Slide17
Guided Practice
Suppose a battleship has mass around 51,000 metric tons(51,000,000
kg) and a k value of about 59,000 kg/sec. Assumethe 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
a
bout 1899.296 seconds, or
in about 31.655 minutesSlide18
Guided Practice
Suppose a battleship has mass around 51,000 metric tons(51,000,000
kg) and a k value of about 59,000 kg/sec. Assumethe 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