Question

Write the exponential function y = 22(1.35)' in the form y = aekt.

Answer 1

Answer:

• y = 22·e^(0.3001t), growing
• annual rate: 35%
• continuous rate: 30.01%

Step-by-step explanation:

When an exponential function is in the form ...

  y = ab^t

the factor 'b' is called the growth factor. When t is in years, it is related to the annual growth rate (r) by ...

  b = 1 +r

In the alternate form, this can be written ...

  y = a·e^(kt)

Comparing the forms, we see that ...

  b = e^k   ⇒   k = ln(b)

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(a)

  y = 22·1.35^t

  k = ln(1.35) ≈ 0.3001

  y = 22·e^(0.3001t)

The growth factor is greater than 1, so the function is growing.

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(b)

The annual growth rate is ...

  r = 1.35 -1 = 0.35 = 35.00%

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(c)

The continuous growth rate is ...

  k = 0.3001 = 30.01%