Suppose the population of trout in a certain stretch of a river is 4000. In about how many years will the population of trout be 700 if the decay rate is 35%? Use the equation [tex]$700=(4000)(0.65)^x$[/tex] and round the value of [tex]$x$[/tex] to the nearest year.
Answer (2)
Divide both sides of the equation by 4000: 4000 700 = ( 0.65 ) x .
Simplify the fraction: 0.175 = ( 0.65 ) x .
Take the natural logarithm of both sides and use the power rule: ln ( 0.175 ) = x ln ( 0.65 ) .
Solve for x and round to the nearest year: x = l n ( 0.65 ) l n ( 0.175 ) ≈ 4 .
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Explanation
Understanding the Problem We are given the equation 700 = ( 4000 ) ( 0.65 ) x , which models the population of trout in a river after x years. We want to find the value of x , rounded to the nearest year.
Isolating the Exponential Term First, we divide both sides of the equation by 4000: 4000 700 = ( 0.65 ) x
Simplifying the Fraction Simplifying the fraction, we get: 0.175 = ( 0.65 ) x
Taking the Natural Logarithm To solve for x , we take the natural logarithm of both sides: ln ( 0.175 ) = ln (( 0.65 ) x )
Applying the Power Rule of Logarithms Using the power rule of logarithms, we have: ln ( 0.175 ) = x ln ( 0.65 )
Solving for x Now, we solve for x by dividing both sides by ln ( 0.65 ) : x = ln ( 0.65 ) ln ( 0.175 )
Calculating the Value of x Calculating the value of x , we find that x ≈ 3.97 .
Rounding to the Nearest Year Rounding x to the nearest whole number, we get x = 4 . Therefore, it will take approximately 4 years for the population of trout to be 700.
Examples
Exponential decay is a concept used in various real-world scenarios, such as calculating the depreciation of a car's value over time. For instance, if a car initially costs 25 , 000 an dd e p rec ia t es a t a r a t eo f 15 V = V_0(1 - r)^t , w h ere V i s t h e f ina l v a l u e , V_0 i s t h e ini t ia l v a l u e , r i s t h e d ec a yr a t e , an d t$ is the time in years. Understanding exponential decay allows us to estimate future values and plan accordingly.
The trout population will decrease from 4000 to 700 in approximately 4 years, given a decay rate of 35%. To find this, we used the equation and simplified it. After taking the natural logarithm and solving for the variable, we rounded the result to get the final answer.
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