SOLUTION: The hourly wage of some Toyota assembly line workers increased annually by the same percentage if their wage went from 7.10 to 20.08 in 12 years what is the doubling time

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Question 1203682: The hourly wage of some Toyota assembly line workers increased annually by the same percentage if their wage went from 7.10 to 20.08 in 12 years what is the doubling time
Found 3 solutions by ikleyn, Edwin McCravy, math_tutor2020:
Answer by ikleyn(52832)   (Show Source): You can put this solution on YOUR website!
.
The hourly wage of some Toyota assembly line workers increased annually by the same percentage
if their wage went from 7.10 to 20.08 in 12 years what is the doubling time
~~~~~~~~~~~~~~~~~~~~~

According to the problem, for the annual salary we have this exponential equation

    y = ,  where "t" denotes years from the starting moment.


We want to find (1+r), the coefficient of growth.
At t= 12 we have

    20.08 = .


Divide both sides by 7.10

     = ,  or

    2.828169014 = .


Take logarithm base 10 of both sides

    log(2.828169014) = 12*log(1+r)

    log(1+r) =  = 0.037625447.


Hence,

    1+r =  = 1.09049944.


Thus we just found out the coefficient of growth.


To find the doubling period, we shoud find "t" from this equation

    2 = ,  or

    2 = .


Again, take logarithm base 10 of both sides

    log(2) = t*log(1+r).


We just found out the value of log(1+r) above: it is  0.037625447.


Therefore, the doubling time is  t =  = 8.000702 years,
which we can round to 8 years.


ANSWER.  The doubling period is 8 years.

Solved.




Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
It more than doubled in 12 years, so it won't take that
long to double.

























Now to double from P to 2P



 



  









Answer: 8 years.

Edwin



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Refer to the steps tutor @Edwin has provided to determine that r = 0.09 = 9% approximately.

Another approach would be to find the positive root of the function
Use a graphing tool like a TI83/TI84, Desmos, or GeoGebra.
The root finding feature quickly locates the positive x intercept to be roughly x = 0.09

Then we can use the rule of 72 shortcut.
We divide 72 over the r value, in which we have removed the percent sign. Do not use the decimal form.

doubling time = 72/(r value without percent sign)
doubling time = 72/9
doubling time = 8 years

This shortcut gives us an approximation rather than the exact answer.
But sometimes all that's needed is a good enough approximation.

----------------------------

A more accurate doubling time formula is

where the 2 in ln(2) refers to doubling.
Replace 2 with 3 if you want to know how long it takes for your money to triple for instance.

If r = 0.09, then we'll get the following doubling time




approximately, which is fairly close to t = 8 we got earlier.

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