HOW TO FIND EXPONENTIAL GROWTH AND DECAY RATE

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%.