A tree grows 2.80 m during the first year since it was planted. During each subsequent year the tree's growth is 85% of its growth the previous year.
a) Calculate to the nearest 0.001m, the growth of the tree in the fourth year.
Tn = ar^n-1
T4 = 2.80(0.85)^10-1
T4 = 2450
I don't get how to do this, the answer is wrong, its supposed to be 1.72m :\
Determine the first year in which the growth of the tree is less than half a metre.
Where in the world did you get 10 in the 10-1
should have been 4-1
T4 = 2.8(.85)^3 = 1.719 or 1.72
for growth to be .5
.5 = 2.8(.85)^(n-1)
divide by 2.8
.17857.. = .85^(n-1)
take log of both sides
log (.17857..) = log [.85^(n-1)]
log (.17857..) = (n-1)log [.85]
n-1 = log (.17857..) / log [.85] = 10.6
n = 11.6
in year 11, growth = 2.8(.85)^10 = .5512
in year 12, growth = 2.8(.85)^11 = .468
so what do you think?
So that would be Year 12.
yes
To calculate the growth of the tree in the fourth year, we can use the formula for the n-th term of a geometric sequence:
T_n = a * r^(n-1)
where:
T_n is the n-th term (in this case, the growth of the tree in the nth year),
a is the initial growth of the tree (2.80 m in this case),
r is the common ratio (0.85 in this case),
and n is the number of years.
For the fourth year, n = 4. Plugging these values into the formula, we get:
T_4 = 2.80 * 0.85^(4-1)
T_4 = 2.80 * 0.85^3
T_4 ≈ 2.80 * 0.614125
T_4 ≈ 1.71835
So, the growth of the tree in the fourth year is approximately 1.718 meters (rounded to the nearest 0.001m), not 1.72m as stated earlier.
Now, let's determine the first year in which the growth of the tree is less than half a meter:
We can set up an inequality:
T_n < 0.5
Substituting the formula for T_n, we get:
2.80 * 0.85^(n-1) < 0.5
Dividing both sides by 2.80:
0.85^(n-1) < 0.5 / 2.80
0.85^(n-1) < 0.178571
To find the first year when the growth is less than half a meter, we need to solve for n. We can take the logarithm of both sides of the inequality to isolate the exponent:
log base 0.85 (0.85^(n-1)) < log base 0.85 (0.178571)
(n-1) * log base 0.85 (0.85) < log base 0.85 (0.178571)
Since log base 0.85 (0.85) is equal to 1, the inequality simplifies to:
(n-1) < log base 0.85 (0.178571)
Now, we can solve for n by adding 1 to both sides:
n < log base 0.85 (0.178571) + 1
Using logarithmic functions or a scientific calculator, we can evaluate this expression to find the smallest value of n for which the inequality holds true.