Question

Use the approximate half-life formula for the case described below. Discuss whether the formula is valid for the case described.Urban encroachment is causing the area of a forest to decline at the rate of 9% per year. What is the half-life of the forest? What fraction of the forest will remain in 30 years?(Type an integer or decimal rounded to the nearest hundredth as needed.)

Answer 1

The approximate half-life formula can be applied to the case of a forest declining due to urban encroachment. The half-life of the forest can be found using the decay factor formula, and the fraction of the forest remaining after a given time can be calculated using the decay factor and the number of half-lives elapsed.

Step-by-step explanation:

The approximate half-life formula is commonly used in radioactive decay, but it can also be applied to other scenarios, such as the decline of a forest due to urban encroachment. The formula states that the amount remaining after a given time is equal to the initial amount multiplied by 0.5 raised to the power of the number of half-lives elapsed.

In this case, the forest is declining at a rate of 9% per year, which is equivalent to a 0.91 decay factor. The decay factor can be found by subtracting the percentage rate of change from 1 (1 - 0.09 = 0.91). The decay factor is then raised to the power of the time elapsed in terms of the half-life.

To find the half-life of the forest, we need to determine how long it takes for the forest to decline to 50% of its original area. We can use the decay factor of 0.91 and the formula mentioned earlier to solve for the half-life.

1. 0.5 = 1 * 0.91t/h
2. 0.5 = 0.91t/h
3. Take the logarithm of both sides using a calculator (base 0.91): log0.910.5 = t/h
4. Approximately t/h = 3.06 half-lives
5. The half-life of the forest is approximately 3.06 years.

To find the fraction of the forest remaining after 30 years, we divide the time elapsed by the half-life:

1. 30 years / 3.06 years = ~ 9.80 half-lives
2. Using the decay factor of 0.91, we can calculate the fraction remaining as: 0.919.80 = ~ 0.145

Therefore, approximately 14.5% of the forest will remain after 30 years.

Answer 2

Half-life = 7.35 years

After 30 years 0.06 of the forest will remain

Explanation:

Half-life is the amount of time it takes the forest to decline to half its initial value.

Now we are told that the forest declines at a rate of 9% per year. This means the amount left next year is 100% - 9% = 91% of the previous. Therefore, if we call the initial amount A, then the amount left after t years will be

`P(t)=A((91\%)/(100\%))^t`
`\Rightarrow P(t)=A(0.91)^t`

Now, when the forest declines to half its initial value, we have

`(A)/(2)=A(0.91)^t`

Canceling A from both sides gives

`(1)/(2)=0.91^t`

Taking the logarithm (of base 0.91) of both sides gives

`\log_(0.91)((1)/(2))=t`
`t=7.35\text{ years.}`