Question

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

(a) Once you have rewritten the formula, give k accurate to at least four decimal places.
k%3D
help (numbers)
If t is measured in years, indicate whether the exponential function is growing or decaying and find the annual and
continuous growth/decay rates. The rates you determine should be positive in the case of growth or decay (by choosing
decay the negative rate is implied).
(b) The annual? rate is
% per year (round to the nearest 0.01%).

Answer 1

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

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)

__

(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.

__

(b)

The annual growth rate is ...

r = 1.35 -1 = 0.35 = 35.00%

__

(c)

The continuous growth rate is ...

k = 0.3001 = 30.01%