c) The equation for N(t) is
N(t) = 400(1.06)^t
recall, the equation for exponential growth is expressed as
y = Po(1 + r)^t
where
r represents the growth rate
By comparing both equations,
1 + r = 1.06
r = 1.06 - 1
r = 0.06
We would convert the rate to percent by multiplying by 100. Thus,
percent growth each year = 0.06 x 100 = 6%
d) The equation for Tasha's balance is
T(t) = 650(1.03)^t
To find the year when Nick's balance would catch up with tasha's, we would equate both equations and solve for t. We have
400(1.06)^t = 650(1.03)^t
Dividing both sides of the equation by 400, it becomes
400/400(1.06)^t = 650/400(1.03)^t
(1.06)^t = 1.625(1.03)^t
1.625 = (1.06)^t/(1.03)^t
Taking the natural log of both sides, it becomes
ln 1.625 = ln[(1.06)^t/(1.03)^t] = ln(1.06)^t - ln(1.03)^t
By applying one of th rules of logarithm, it becomes
ln 1.625 = tln1.06 - tln1.03 = t(ln1.06 - ln1.03)
t = ln 1.625/(ln1.06 - ln1.03)
t = 16.9
It will take approximately 16.9 years for Nick's balance to catch up with Tasha's
The graph is shown below
The red line represents Nick's balance after t years
the blue line represents Tasha's balance after t years