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Question 1129689: Suppose f is an exponential function and:
b represents the 1-unit growth factor for f.
c represents the n-unit growth factor for f.
d represents the m -unit growth factor for f.
A)Write a formula that expresses c in terms of b.
B)Write a formula that expresses b in terms of c.
C)Write a formula that expresses d in terms of c.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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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