**Exponential Growth and Decay**

**Sol 1** Since , we have that
.

Since when , ; so . Taking natural logarithms of both sides gives , so .

Substituting back in the formula for gives

, so

.

Taking natural logarithms on both sides gives

, so and

**Sol 2** Since , we have that
.

Since when , ; so . Taking natural logarithms of both sides gives ; so .

Substituting back in the formula for gives

.

To find the half-life, we can set and solve for :

.

Taking natural logarithms on both sides gives

, so and

**Sol 3** We know that , so
.

Since when , we have that

and therefore .

Taking natural logarithms on both sides yields

, so and therefore

.

When the population has increased by 40%, it will be equal to ; so

.

Taking natural logarithms gives

, so and hence

**Sol 4** Since ,
. To find the time required for
the amount to triple, we can set and then solve for :

, so

years.

**Sol 5** Here , so
.

Since when , we have that , so

.

Thus .

The number of words remembered has decreased by 40% when , so

;

and taking natural logarithms on both sides gives

. Therefore we get that

**Sol 6** Since the number of people infected doubles every 5 weeks, and since
, the number of people infected will increase by a factor of 8 [that is,
double 3 times] in weeks.

**Sol 7** Let be the number of grasshoppers in the town (in thousands)
after t days.

Then we know that

since the proportional amount of change in any 15-day time period is the same, so

and therefore

thousand grasshoppers.

**Sol 8A** Since when and when , we have that

and .

Dividing the second equation by the first gives

so . Taking the square root of both sides gives

, so substituting back in the first equation gives

, so

bugs.

**Sol 8B** Making a time-shift, let correspond to the time when there
were 2880 bugs.

Then where , so ; and when .

Therefore , so .

Taking square roots of both sides gives , so substituting back in the formula for gives

.

Since we shifted the time by 3 weeks, the initial number of bugs is given by

.

**Sol 9** For state A, we have that (in millions) where

when ; so letting in the formula for gives

.

For state B, we have that (in millions) where

when ; so letting in the formula for gives

.

Setting the expressions for the populations of the two states equal to each other and solving for , we get

and

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