The Forever Green Nursery owns 7000 white pine trees. Each year the nursery plans to sell 12% of the trees and to then plant 600 new ones.

a. Find the # of trees owned by the nursery after 10 year
b. Find the # of trees owned by nursery after many years and explain what is occurring
c. What equation can you solve to find the number of trees in the long run
d. Try different starting totals in place of the 7000 trees. Describe any changes to long-run value
e. In the fifth year, a disease destroys many of the nursery's trees. How does the long-run value change?

x = number of trees at year n

at year 0, x0 = 7000

at year 1, x1 =.88*X0 + 600 = 6160+600 = 6760

at year 2, x2 = .88*x1 +600 = 6548.8

at year 3 , x3 = .88x2+600 = 6362.9

at year 4, x4 = .88*x3 + 600 = 6199.4

at year 5, x5 = .88*x4+600 = 6055.5

at year 6, x6 = .88*x5+600 = 5928.8

at year 7, x7 = .88*x6+600 = 5817.3

at year 8, x8 = .88*x7+600 = 5719.3

at year 9, x9 = .88*x8+600 = 5633

at year 10,x10= .88*x9+600 = 5557 which is answer to a.

b.
when .12 xn = 600
we lose as many as we gain
xn = 600/.12 = 5000
notice above that we are approaching 5000 as we get to year ten
well, I did part c to explain part b

d. and e.
sure go ahead, but .12 xn = 600 always wins in the long run

a. To find the number of trees owned by the nursery after 10 years, we need to calculate the changes in the tree count each year.

After the first year, the nursery will sell 12% of the trees: 0.12 * 7000 = 840 trees.
They will then plant 600 new trees: 840 - 600 = 240 trees.

So after the first year, the nursery will have 7000 - 840 + 240 = 6160 trees.

In the second year, they will again sell 12% of the trees: 0.12 * 6160 = 739.2 trees (rounded to 739 trees).
They will plant 600 new trees: 739 - 600 = 139 trees.

After the second year, the nursery will have 6160 - 739 + 139 = 5470 trees.

Continuing this pattern for 10 years, we find that the nursery will have approximately 4888 trees after 10 years.

b. After many years, the number of trees owned by the nursery will approach a certain value. This value occurs when the number of trees selling each year equals the number of trees planted. In this case, that number is 600.

c. The equation to solve for the number of trees in the long run can be expressed as:

Number of trees sold each year = Number of trees planted each year.

Let's call the long-run value L. With the given information, we can set up the equation:

0.12L = 600

Simplifying:

L = 600 / 0.12
L = 5000

Therefore, in the long run, the nursery will have 5000 trees.

d. If we try different starting totals for the number of trees, it will eventually converge to the long-run value of 5000 trees. However, the time it takes to reach that value may vary. A higher starting total will result in more trees initially, but the difference will reduce over time as the long-run value is approached.

e. If a disease destroys many of the nursery's trees in the fifth year, the long-run value will still remain unchanged. The disease only affects the immediate count, but as long as the nursery continues to sell 12% of the trees and plant 600 new ones each year, the long-run value will remain at 5000 trees. The diseased trees will be replaced by the new ones planted, thus maintaining the overall balance.

a. To find the number of trees owned by the nursery after 10 years, we can use the following steps:

1. Calculate the number of trees sold each year: 12% of 7000 = 0.12 * 7000 = 840 trees.
2. Calculate the number of trees planted each year: 600 trees.
3. Calculate the number of trees remaining after 10 years: 7000 - (10 * 840) + (10 * 600) = 7000 - 8400 + 6000 = 4600 trees.

Therefore, the nursery will own 4600 trees after 10 years.

b. In the long run, the number of trees owned by the nursery will eventually stabilize. This occurs because each year, the number of trees sold is offset by the number of trees planted. As a result, the overall tree count remains relatively constant.

c. We can represent the number of trees in the long run using the equation:
Number of trees = Starting number of trees - (Rate of trees sold * Starting number of trees) + (Rate of trees planted * Time)

d. When trying different starting totals, the long-run value will change depending on the initial number of trees. For example, if we start with a higher number of trees, the long-run value will be higher. Conversely, if we start with a lower number of trees, the long-run value will be lower. This is because the starting total directly affects the number of trees sold each year, which then impacts the final count of trees in the long run.

e. If in the fifth year, a disease destroys many of the nursery's trees, it will directly impact the number of trees remaining. The disease will reduce the number of trees, which might affect the long-run value. Depending on the severity of the disease and the number of trees affected, the long-run value could decrease significantly. To determine the exact change, further information about the number of trees destroyed is needed.

a. To find the number of trees owned by the nursery after 10 years, we need to understand the annual changes in the tree count.

Each year, the nursery plans to sell 12% of the trees and plant 600 new ones.

The number of trees sold each year can be calculated by multiplying 12% by the total number of trees owned. So, the number of trees sold each year would be 0.12 * 7000 = 840.

After selling the trees, the number of trees that remain would be 7000 - 840 = 6160.

Adding the 600 newly planted trees, the total number of trees after the first year would be 6160 + 600 = 6760.

For the subsequent years, we repeat the process. Selling 12% of the trees each year and adding 600 new trees.

So, after 10 years, the number of trees owned by the nursery would be:
6760 - (0.12 * 6760) + 600)^(10) ≈ 5550.92

Therefore, the nursery would own approximately 5551 trees after 10 years.

b. To find the number of trees owned by the nursery after many years, we can use the concept of a recursive formula.

Let's assume the number of trees owned by the nursery after year n is represented by T(n).

For each year, the number of trees owned would be:
T(n) = T(n-1) - (0.12 * T(n-1)) + 600

By plugging in the value of T(0) (initial number of trees), we can recursively calculate the number of trees owned by the nursery for subsequent years.

As the number of years increases, the number of trees owned by the nursery will gradually approach a stable value known as the long-run value.

c. To find the number of trees in the long run, we can set the recursive formula equal to the value in the long run and solve for T.

T(n) = T(n-1) - (0.12 * T(n-1)) + 600
T(n) = T(n-1) * 0.88 + 600

In the long run, the tree count remains constant, so we can set T(n) = T(n-1) and solve for T.

T = T * 0.88 + 600
T = T * 0.88
0.12T = 600
T = 600 / 0.12
T ≈ 5000

Therefore, the number of trees in the long run would be approximately 5000.

d. If we try different starting totals instead of the initial 7000 trees, the long-run value will still be approximately 5000. This is because the annual changes (selling 12% and planting 600) gradually reach a stable equilibrium where the number of trees sold is balanced by the number of newly planted trees.

For example, if we start with 5000 trees, after many years, the number of trees owned by the nursery will approach 5000.

Similarly, if we start with 10,000 trees, after many years, the number of trees owned by the nursery will also approach 5000.

Therefore, the long-run value remains consistent regardless of the initial starting total.

e. In the fifth year, if a disease destroys many of the nursery's trees, the long-run value will be affected. The impact depends on the severity of the disease and the number of trees affected.

If the disease significantly reduces the number of trees, the long-run value will be lower than the approximate 5000. This is because the reduced number of trees will limit the potential for growth and balance between sales and new plantings.

To determine the exact impact on the long-run value, we would need more information about the disease and its effect on the tree count.