How to find 1 unit growth factor

OK, a "growth factor" is just "What do you multiply your count at time t by, to get the count at time t+1?" It's usually expressed as 1 + a percentage; so, if things are growing at 5% per year, you'd usually express the growth factor as 1.05. This may or may not be the case with your course/teacher/textbook/website/whatever, so check that out to be sure.

OK, so you're told the number triples every 5 hours. Question b. wants you to make equations based on b to show what the number is at 11, 12, 13, 14, and 15 hours. Well, let's call the number at time t n*t. So n10* = 36, and n*15* = 108.

The growth factor b means that n*t+1* = n*t* ∙ b. That should help you put the right answers in for 11, 12, 13, 14, 15. The trick might be something like this, though:

• Example of n*13: Do they want the answer n12* ∙ b, or do they want the answer n*11* ∙ b2, or do the want the answer n*10* ∙ b3?

In c., they want you to take the n*15* = 108 that you have from a., and the n*15* = [something involving b] from b. Then set them equal to each other, and solve for b.

In d., that should be pretty easy, but don't just do it on your calculator to give an approximate number; use exponents or whatever to give the proper exact answer.

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(I) The growth factor is the ratio of any two successive measurements.

The growth factor is 1/6.

(II) The value changes from 1 to 1/6, a decrease of 5/6. The fractional change is -5/6; the percent change is - 83 1/3%.

(III) Given: the initial value (when x=0) is 21.6.

(IV) F(x) = 21.6(1/6)^x

Note: Use the fractional form of the growth factor because it is exact; a decimal would only be an approximation.

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the basic formula for exponential growth is f = p * (1 + r) ^ n

f is the future value
p is the prewent value
r is the growth rate per time period
n is the number of time periods.

let the time periods be number of units.

for 1 unit of growth, the formula becomes f = p * (1 + r) ^ 1
for n units of growth, the formula becomes f = p * (1 + r) ^ n
for m units of growth, the formula becomes f = p * (1 + r) ^ m

if you divide both sides of each euation by p, then you get:

f/p = (1 + r) ^ 1
f/p = (1 + r) ^ n
f/p = (1 + r) ^ m

(1 + r) ^ 1 is the 1-unit growth factor.
(1 + r) ^ n is the n-units growth factor.
(1 + r) ^ m is the m-units growth factor.

since b represents the 1-unit growth factor, you get b = (1 + r) ^ 1
since c represents the n-units growth factor, you get c = (1 + r) ^ n
since d represents the m-units growth factor, you get d = (1 + r) ^ m

c = (1 + r) ^ n can be shown as c = (1 + r) ^ (1 * n)
this can be shown as c = ((1 + r) ^ 1) ^ n.
since b = (1 + r) ^ 1, then:
c = ((1 + r) ^ 1) ^ n can be shown as c = b ^ n.

b = (1 + r) ^ 1 can be shown as b = (1 + r) ^ (n * 1/n).
this can be shown as b = ((1 + r) ^ n) ^ (1/n).
since c = (1 + r) ^ n), then:
b = ((1 + r) ^ n) ^ (1/n) can be shown as b = c ^ (1/n).

c = (1 + r) ^ n
d = (1 + r) ^ m

d = (1 + r) ^ m can be shown as d = (1 + r) ^ (n * k), where k = m/n.
this can be shown as d = ((1 + r) ^ n) ^ k.
since c = (1 + r) ^ n), then:
d = ((1 + r) ^ n) ^ k can be shown as d = c ^ k
since k = m / n, then:
d = c ^ k can be shown as d = c ^ (m / n)

one method to confirm that these formulas are correct is to take random values of m and n and plugging them into the formulas to see if the formulas hold true.

the formulas involved are:

b = (1 + r) ^ 1 which can also be shown as b = c ^ (1/n)
c = (1 + r) ^ n which can also be shown as c = b ^ n
d = (1 + r) ^ m which can also be shown as d = c ^ (m / n)

if we let n = 15 and m = 32 and we let r = .07 (chosen randomly but keeping them small enough so the calculations don't get ridiculously large), you get:

b = (1 + r) ^ 1 becomes b = 1.07 ^ 1 = 1.07
c = (1 + r) ^ n becomes c = 1.07 ^ 15 = 2.759031541
d = (1 + r) ^ m becomes c = 1.07 ^ 32 = 8.715270798

b = c ^ (1/n) becomes b = 2.759031541 ^ (1/15) = 1.07, which is true.

c = b ^ n becomes c = 1.07 ^ 15 = 2.759031541, which is true.

d = c ^ (m / n ) becomes d = 2.759031541 ^ (32 / 15) = 8.715270798, which is true.

the formula are good.

your solutions are:

b = c ^ (1/n)
c = b ^ n
d = c ^ (m / n)

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