<< Chapter < Page Chapter >> Page >

Given the basic exponential growth    equation A = A 0 e k t , doubling time can be found by solving for when the original quantity has doubled, that is, by solving 2 A 0 = A 0 e k t .

The formula is derived as follows:

2 A 0 = A 0 e k t 2 = e k t Divide by  A 0 . ln 2 = k t Take the natural logarithm . t = ln 2 k Divide by the coefficient of  t .

Thus the doubling time is

t = ln 2 k

Finding a function that describes exponential growth

According to Moore’s Law, the doubling time for the number of transistors that can be put on a computer chip is approximately two years. Give a function that describes this behavior.

The formula is derived as follows:

t = ln 2 k The doubling time formula . 2 = ln 2 k Use a doubling time of two years . k = ln 2 2 Multiply by  k  and divide by 2 . A = A 0 e ln 2 2 t Substitute  k  into the continuous growth formula .

The function is A = A 0 e ln 2 2 t .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Recent data suggests that, as of 2013, the rate of growth predicted by Moore’s Law no longer holds. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account.

f ( t ) = A 0 e ln 2 3 t

Got questions? Get instant answers now!

Using newton’s law of cooling

Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the object’s temperature will decrease exponentially, leveling off as it approaches the surrounding air temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless the room temperature is zero, this will correspond to a vertical shift    of the generic exponential decay function. This translation leads to Newton’s Law of Cooling    , the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature

T ( t ) = a e k t + T s

This formula is derived as follows:

T ( t ) = A b c t + T s T ( t ) = A e ln ( b c t ) + T s Laws of logarithms . T ( t ) = A e c t ln b + T s Laws of logarithms . T ( t ) = A e k t + T s Rename the constant  c   ln   b ,  calling it  k .

Newton’s law of cooling

The temperature of an object, T , in surrounding air with temperature T s will behave according to the formula

T ( t ) = A e k t + T s
  • t is time
  • A is the difference between the initial temperature of the object and the surroundings
  • k is a constant, the continuous rate of cooling of the object

Given a set of conditions, apply Newton’s Law of Cooling.

  1. Set T s equal to the y -coordinate of the horizontal asymptote (usually the ambient temperature).
  2. Substitute the given values into the continuous growth formula T ( t ) = A e k t + T s to find the parameters A and k .
  3. Substitute in the desired time to find the temperature or the desired temperature to find the time.

Using newton’s law of cooling

A cheesecake is taken out of the oven with an ideal internal temperature of 165°F, and is placed into a 35°F refrigerator. After 10 minutes, the cheesecake has cooled to 150°F . If we must wait until the cheesecake has cooled to 70°F before we eat it, how long will we have to wait?

Because the surrounding air temperature in the refrigerator is 35 degrees, the cheesecake’s temperature will decay exponentially toward 35, following the equation

T ( t ) = A e k t + 35

We know the initial temperature was 165, so T ( 0 ) = 1 6 5 .

165 = A e k 0 + 35 Substitute  ( 0 , 165 ) . A = 130 Solve for  A .

We were given another data point, T ( 1 0 ) = 1 5 0 , which we can use to solve for k .

                150 = 130 e k 10 + 35 Substitute (10, 150) .                 115 = 130 e k 10 Subtract 35 .                115 130 = e 10 k Divide by 130 .           ln ( 115 130 ) = 10 k Take the natural log of both sides .                      k = ln ( 115 130 ) 10 = 0.0123 Divide by the coefficient of  k .

This gives us the equation for the cooling of the cheesecake: T ( t ) = 1 3 0 e 0 . 0 1 2 3 t + 3 5 .

Now we can solve for the time it will take for the temperature to cool to 70 degrees.

70 = 130 e 0.0123 t + 35 Substitute in 70 for  T ( t ) . 35 = 130 e 0.0123 t Subtract 35 . 35 130 = e 0.0123 t Divide by 130 . ln ( 35 130 ) = 0.0123 t Take the natural log of both sides t = ln ( 35 130 ) 0.0123 106.68 Divide by the coefficient of  t .

It will take about 107 minutes, or one hour and 47 minutes, for the cheesecake to cool to 70°F .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Please prove it
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
the 28th term is 175
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?