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Trees of the same age, size and type are growing at the same rate in a forest - Leaving Cert Mathematics - Question b - 2021
Question b
Trees of the same age, size and type are growing at the same rate in a forest. It is possible to measure the radius of the trunk of a tree at the end of each growing... show full transcript
Worked Solution & Example Answer:Trees of the same age, size and type are growing at the same rate in a forest - Leaving Cert Mathematics - Question b - 2021
Step 1
Show that r(t) satisfies the condition.
Answer
To show that r(t) is increasing, we need to demonstrate that r(t + 1) > r(t).
Starting from the formula:
r(t + 1) = r(t) + i(t + 1)
Here, since i(t) is the increase in radius, as long as i(t + 1) > 0, it follows that:
r(t + 1) > r(t)
Given
i(t) = 1.5 + sin ( \frac{\pi t}{5} )
The sine function oscillates between -1 and 1. Therefore:
0 < i(t) < 2.5
This ensures that the radius is increasing each year, satisfying the condition of increasing radius.
Step 2
Show that i(6) < i(5) and explain what this means in the context of the growth of a tree.
Answer
First, we calculate:
i(5) = 1.5 + sin ( \frac{\pi imes 5}{5} ) = 1.5 + sin(\pi) = 1.5 + 0 = 1.5
i(6) = 1.5 + sin ( \frac{\pi imes 6}{5} ) = 1.5 + sin(\frac{6\pi}{5})
Since sin(\frac{6\pi}{5}) < 0, i(6) < 1.5, so we conclude: i(6) < i(5).
This means that the increase in the radius of the tree at year 6 is less than that of year 5, indicating a slowing growth rate.
Step 3
Find r(2), the radius of the second tree at t = 2.
Answer
Using the formula:
r(t) = r(0) + ( \sum_{k=1}^{t} i(k) )
We calculate:
r(1) = r(0) + i(1), r(2) = r(1) + i(2).
Substituting: r(1) = 10 + (1.5 + sin ( \frac{\pi imes 1}{5} )) = 10 + (1.5 + 0.309) = 11.809
Now find r(2): r(2) = r(1) + i(2) r(2) = 11.809 + (1.5 + sin ( \frac{2 \pi}{5} )) = 11.809 + (1.5 + 0.951) = 13.260
Thus, r(2) = 13 + sin ( \frac{2\pi}{5} ) + sin ( \frac{2\pi}{5} ) where a = 13, b = 2, and c = 2.
Step 4
Find the value of k.
Answer
At t = 10, we have: r(10) = 10 + ( \sum_{k=1}^{10} i(k) )
Using i(t), we compute: r(10) = 10 + 10(1.5 + sin ( \frac{10\pi}{5} )) = 10 + 10(1.5 + 0) = 25.
The volume of the trunk cut is: V₂ = kV₁ → V₁ is calculated using: V₁ = ( r(10) \times h \pi )
This simplifies the calculation for k, k = ( \frac{V₂}{V₁} ). Thus, k = ( \frac{25h \pi}{h \pi} = 25 ).