Solving Exponential Functions/Applications
I have several questions, I'd appreciate if anyone would give tips/solutions/corrections!
1) solve : 32x -5(3x)=-6
answer: 1 & 0.631
I tried to divide out the "-5", so the right side was "6/5". Then I converted it to log forms and solved for x by dividing them, however, this didn't get me the right value.
2) The half-life of stronium-90 is 25 years. For 100 mg sample, estimate the rate of decay after 15 years.
answer: -1.83 mg/yr
I know that the formula is M=Mo(0.5)t/hl and I subbed in the values.
M = (100)(0.5)15/25
M = 65.98 mg
Now I'm not sure how to find the rate of decay? Any suggestions?
3) The population of a small town increases by 3.5% per year. If the population in 1995 was 2300 people, estimate the rate of growth of the population in 2005.
answer: approx. 112 people/year
I subbed in the values into y = c(1+i)x and the population in 2005 will be about 3244 people. Again, not sure how to find the rate of growth.
4) Stronium-90 loses 2.5% of its mass each year. What is its half life? Estimate the rate at which he mass is decreasing after 5 years.
answer one: 27.38 years
answer two: 2.23%/year
Now I'm not sure how to solve this one because I feel like its missing some values and I don't know how to work with the formulas then.
Thank you in advance!
Re: Solving Exponential Functions/Applications
Hint Problem 1: Try and factor out the 3^x term and get a quadratic. For problem 3^(2x) -5(3^[x])=-6 use a substitution y = 3^x and you obtain y^2 - 5y + 6 = 0 and you can use the
quadratic equation to find a solution.
Hint Problem 2: Consider differentiating the function with respect to time.
Hint Problem 3: You used the sum of a geometric series and in R I got the following output:
For the rate of growth you have been given the information and depending on how you want to specify the variable, it can either be r = 1 + 0.0035 or r = 0.0035. In a geometric series formulation its r = 0.0035 and in another formulation its 1.0035.
Hint Problem 4: Assuming the problem is dealing with discrete differentials (geometric series) then consider the half life as the rate at which something decays every cycle (i.e. per period). You said that that the mass loss is 2.5% so this corresponds to the radioactive decay since the mass is proportional the atomic weight which represents a configuration of elementary particles (including protons, neutrons, and electrons). (I should point out, they are looking for unknowns currently that may appear in the LHC and other planned similar experiments).
Also please note, if you specify continuous rates of change you need to use the exponential formulation and if its discrete you need to use the series formulation (geometric series). If you need to use continuous then take a look at the exponential.