Question

Piney woods conservation is a company that attempts to help offset the effects of deforestation. a local forest contains approximately 500,000 trees. lumber companies are continuously clearing the forest at a rate of 4.7% per year. piney woods conservation is about to begin planting trees in the region throughout each year at an average rate of 15,000 trees per year. they are curious to know how long it will be before the number of trees they have planted will be equal to the number of trees still remaining in the forest. create a system of equations to model the situation above, and use it to determine if there are any solutions. if there are any solutions, determine if they are viable or not.

Answer 1

To determine when reforestation efforts will equal the remaining trees after deforestation, model the deforestation with an exponential decay equation T(t) = 500,000 × (1 - 0.047)^t and reforestation with a linear growth equation P(t) = 15,000t, then solve for t when T(t) = P(t).

Step-by-step explanation:

Modeling Deforestation and Tree Planting with Systems of Equations

The situation presented can be modeled by a system of two equations representing the number of trees over time: one for the deforestation by the lumber companies and one for the reforestation efforts by Piney Woods Conservation. Let's denote the initial number of trees by T0 and the time in years as t.

The equation for the deforestation process, considering a continuous decline at a rate of 4.7% per year, would be:

T(t) = T0 × (1 - 0.047)^t

Here, T0 is 500,000, which is the initial number of trees in the forest.

Meanwhile, the equation for the tree planting by Piney Woods Conservation, given they plant 15,000 trees annually, would be:

P(t) = 15,000t

To find the time when both quantities are equal, we set T(t) equal to P(t):

500,000 × (1 - 0.047)^t = 15,000t

This equation can be solved for t using numerical methods or graphically to find when the number of trees planted equals the number of remaining trees.

Analyze the solutions for their viability to determine if and when the reforestation efforts will balance out the deforestation rate. This requires knowledge in algebra and potentially use of a graphing calculator or software.