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