In an over-fished area, the catch of a certain fish is decreasing exponentially.? Use k=0.084 to determine how long will it take for the catch to reach half of its current amount?

The answer is 8.25 years

The exponential function can be expressed as:

A = P * e^ (kt)

A - the final amount

P - the current amount

k - the rate

t - time in years

If the final amount is the half of its current amount, then:

A = P/2

So,

A = P * e^ (kt)

P/2 = P * e^ (kt)

Divide P from both sides:

1/2 = e^ (kt)

Logarithm both sides with natural logarithm:

ln (1/2) = ln (e^ (kt))

ln (0.5) = kt * ln (e)

k = - 0.084

-0.693 = - 0.084 * t * 1

-0.693 = - 0.084t

t = - 0.693 / - 0.084

t = 8.25 years