Question

2. (5 points) The size of the population, P, of toads t years after it isintroduced in a wetlands can be modeled byp=1000/1+49(1/2)^t. How longdoes it take the toad population to reach 750 toads?

Answer 1

Given the function:

`P=(1000)/(1+49((1)/(2))^t)`

Where:

• P is the size of the population of toads

,

• t is the number of years after it is intriduced in a wetlands.

Let's find how long it will take the population to reach 750 toads.

To find the time, substitute 750 for P and solve for t.

We have:

`\begin{gathered} 750=(1000)/(1+49((1)/(2))^t) \\ \\ 750(1+49((1)/(2))^t)=1000 \end{gathered}`

• Divide both sides by 750:

`\begin{gathered} (750(1+49((1)/(2))^t))/(750)=(1000)/(750) \\ \\ 1+49((1)/(2))^t=1.333 \end{gathered}`

• Subtract 1 from both sides:

`\begin{gathered} 1-1+49((1)/(2))^t=1.333-1 \\ \\ 49((1)/(2))^t=0.333 \end{gathered}`

• Divide both sides by 49:

`\begin{gathered} (49((1)/(2))^t)/(49)=(0.333)/(49) \\ \\ ((1)/(2))^t=0.0067959 \\ \\ \end{gathered}`

• Solving further, we have:

`\begin{gathered} (1^t)/(2^t)=0.0067959 \\ \\ (1)/(2^t)=0.0067959 \\ \\ 0.0067959*2^t=1 \\ \\ 2^t=(1)/(0.0067959) \\ \\ 2^t=147.147 \end{gathered}`

• Take the natural loagrithm of both sides:

`\begin{gathered} t\ln (2)=\ln (147.147) \\ \\ t=(\ln (147.147))/(\ln (2)) \\ \\ t=7.2\approx7 \end{gathered}`

Therefore, it will take the toad population approximately 7 years to reach 750 toads.

ANSWER:

7 years