Exponential growth occurs when a quantity increases by the same factor over equal intervals of time.
y = a(1 + r)t
y = final amount
a = Initial amount
r = Rate of growth in decimal form
t = time
(1 + r) = Growth factor
Exponential decay occurs when a quantity increases by the same factor over equal intervals of time.
y = a(1 - r)t
y = final amount
a = Initial amount
r = Rate of decay in decimal form
t = time
(1 - r) = decay factor
Example :
Rewrite the function y = 120(1.25)t/12 to determine whether it represents exponential growth or exponential decay. Then find the percent of change.
Solution :
Write the given function ,
y = 120(1.25)t/12
Using a power property,
= [120(1.25)1/12]t
Evaluate the power
= 120(1.02)t
Rewrite in the form of y = a(1 + r)t.
= 120(1 + 0.02)t
So, the function represents exponential growth and the growth rate is about 0.02 or 2%.
Rewrite the function to
determine whether it represents exponential growth or exponential decay. Then
find the percent of change.
Problem 1 :
y = 80(0.85)2t
Solution :
Write the given function,
y = 80(0.85)2t
Using a power property,
= 80[(0.85)2]t
Evaluate the power
= 80(0.72)t
Rewrite in the form of y = a(1 - r)t.
y = 80(1 – 0.28)t
So, the function represents exponential decay and the decay rate is about 0.28 or 28%.
Problem 2 :
y = 67(1.13)t/4
Solution :
Write the given function,
y = 67(1.13)t/4
Using a power property,
y = 67[(1.13)1/4]t
Evaluate the power
= 67(1.03)t
Rewrite in the form of y = a(1 + r)t.
y = 67(1 + 0.03)t
So, the function represents exponential growth and the growth rate is about 0.03 or 3%.
Problem 3 :
y = 5(3/2)-8t
Solution :
Write the given function,
y = 5(3/2)-8t
Using a power property,
y = 5[(3/2)-8] t
Evaluate the power
= 5(0.04)t
Rewrite in the form of y = a(1 - r)t.
y = 5(1 – 0.96)t
So, the function represents exponential decay and the decay rate is about 0.96 or 96%.
Problem 4 :
y = 17(2/5)0.65t
Solution :
Write the given function,
y = 17(2/5)0.65t
Using a power property,
y = 17[(2/5)0.65]t
Evaluate the power
= 17(0.55)t
Rewrite in the form of y = a(1 - r)t.
y = 17(1 – 0.45)t
So, the function represents exponential decay and the decay rate is about 0.45 or 45%.
Problem 5 :
y = 4(0.5)t/88
Solution :
Write the given function,
y = 4(0.5)t/88
Using a power property,
y = 4[(0.5)1/88]t
Evaluate the power
= 4(0.99)t
Rewrite in the form of y = a(1 - r)t.
y = 4(1 – 0.01)t
So, the function represents exponential decay and the decay rate is about 0.01 or 1%.
Problem 6 :
y = 31(1.02)4t
Solution :
Write the given function,
y = 31(1.02)4t
Using a power property,
= 31[(1.02)4] t
Evaluate the power
= 31[1.08]t
Rewrite in the form of y = a(1 + r)t.
y = 31(1 + 0.08)t
So, the function represents exponential growth and the growth rate is about 0.08 or 8%.
Problem 7 :
y = 9(1.12)0.3t
Solution :
Write the given function,
y = 9(1.12)0.3t
Using a power property,
Evaluate the power
= 9[1.03]t
Rewrite in the form of y = a(1 + r)t.
y = 9(1 + 0.03)t
So, the function represents exponential growth and the growth rate is about 0.03 or 3%.
= 9[(1.12)0.3] t
Evaluate the power
= 9[1.03]t
Rewrite in the form of y = a(1 + r)t.
y = 9(1 + 0.03)t
So, the function represents exponential growth and the growth rate is about 0.03 or 3%.
Problem 8 :
y = 750(0.88)t/3
Solution :
Write the given function,
y = 750(0.88)t/3
Using a power property,
= 750[(0.88)1/3]t
Evaluate the power
= 750(0.96)t
Rewrite in the form of y = a(1 - r)t.
y = 750(1 – 0.04)t
So, the function represents exponential decay and the decay rate is about 0.04 or 4%.
Problem 9 :
y = (0.64)5t
Solution :
Write the given function,
y = (0.64)5t
Using a power property,
= [(0.64)5] t
Evaluate the power
= (0.11)t
Rewrite in the form of y = a(1 - r)t.
y = (1 – 0.89)t
So, the function represents exponential decay and the decay rate is about 0.89 or 89%.
Problem 10 :
y = 6(0.82)-0.25t
Solution :
Write the given function,
y = 6(0.82)-0.25t
Using a power property,
= 6[(0.82)-0.25]t
Evaluate the power
= 6[1.05]t
Rewrite in the form of y = a(1 + r)t.
y = 6(1 + 0.05)t
So, the function represents exponential growth and the growth rate is about 0.05 or 5%.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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