QuestionFind the inequality for when trees drop to 621,859, given 691,255 trees decreasing at $1.5\%$ yearly. Solve for $t$ .

Studdy Solution

STEP 1

Assumptions1. The initial number of trees is691,255. The rate of decrease is1.5% per year3. We are looking for the time (t) when the number of trees will be no more than621,8594. The decrease in the number of trees is a continuous process, not a discrete one

STEP 2

The number of trees in the forest after t years can be modeled by the equation $(t) = N0 (1 - r)^t$ where- $(t)$ is the number of trees after t years, - $0$ is the initial number of trees, - $r$ is the rate of decrease, and- $t$ is the time in years.

STEP 3

We are looking for the time when the number of trees will be no more than621,859. This can be represented by the inequality $(t) \leq621,859$

STEP 4

Substitute the equation from2 into the inequality from3 $0 (1 - r)^t \leq621,859$

STEP 5

Now, plug in the given values for $0$ and $r$ to form the inequality $691,255 (1 -0.015)^t \leq621,859$

STEP 6

To solve for $t$ , we first isolate the term with $t$ on one side of the inequality. We do this by dividing both sides of the inequality by691,255 $(1 -0.015)^t \leq \frac{621,859}{691,255}$

STEP 7

Next, take the natural logarithm of both sides to get rid of the exponent $t \cdot \ln(1 -0.015) \leq \ln\left(\frac{621,859}{691,255}\right)$

STEP 8

Finally, solve for $t$ by dividing both sides by $\ln(1 -0.015)$ $t \geq \frac{\ln\left(\frac{621,859}{691,255}\right)}{\ln(1 -0.015)}$ The inequality sign flips because $\ln(1 -0.015)$ is negative.

STEP 9

Calculate the value of $t$ $t \geq \frac{\ln\left(\frac{621,859}{691,255}\right)}{\ln( -.015)} \approx8.5$ Therefore, it will take approximately8.5 years for the number of trees to decrease to no more than621,859.

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