Question

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?

Answer 1

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