Answered

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9. V = 300(1.02)* models exponential
growth, where t represents the number
of years.
a) What is the annual rate of growth?

b) Calculate the monthly rate of growth
to the nearest hundredth of a percent.

Sagot :

sqdancefan

Answer:

  a) 2%

  b) 0.17%

Step-by-step explanation:

Exponential growth is modeled by the equation ...

  y = a·b^x

where 'a' is the initial value, and 'b' is the growth factor.

When x is has units of time, the growth factor 'b' applies over 1 unit of x. The applicable growth factor for different units of time can be found using the rules of exponents applied to the equation written with x expressed in the different units.

The growth rate is related to the growth factor by ...

  b = 1 +r . . . . where r is the growth rate

  r = b -1 . . . . solved for r

__

a)

Your equation is ...

  V = 300(1.02^t)

where t is expressed in years. Then the corresponding annual growth rate is ...

  b = 1.02

  r = b -1

  r = 1.02 -1 = 0.02 = 2% . . . annual rate of growth

__

b)

When the equation is written so that t is expressed in months, it becomes ...

  V = 300(1.02)^(t/12)

  V = 300(1.02^(1/12))^t

Now, the growth factor (per month) is

  b = 1.02^(1/12)

and the monthly growth rate is ...

  r = 1.02^(1/12) -1 ≈ 1.00165158 -1 = 0.00165158

  r ≈ 0.17% . . . . monthly rate of growth

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