Financial and Calculus problems Watch

smileatyourself
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Below are some questions on financial calculations and calculus for a job interview.

Alas I was not successful but have not been given any feedback from the company.

Maybe I answered some of the questions incorrectly, so I would like to ask readers here to help.



In the early life of a certain kind of tree, it is observed to grow at an exponential rate, whereby at the end of every day it is a fixed fraction taller than it was at the end of the previous day. It is observed that at 1 year it is 1m tall and at 5 years it is 5m tall.

1. How tall was the tree at 2 years?

2. What is the daily, fractional growth rate?

3. How long does it take to double in height?

As the tree grows, it keeps the same proportions. i.e. it grows in width and depth at the same rate as it grows in height.

4. How long does it take to double in weight?

When studied over a longer period of time, it is noticed that the fractional growth rate decreases as the tree gets older. It is suggested that the height h may follow the relationship h = k/(1+e^(-r(t+t0))) for some parameters k, r and t0 and where the tree is planted at time t=0.

5. Assuming this model, what is the maximum height of the tree, in terms of k and r?

6. Suppose a tree can grow to a maximum height of 100 times the height when it is planted. Find a formula for r in terms of t0.

7. At what point in time (again, in terms of k, r and t0) is the tree growing at its maximum rate?

In a managed forest, the goal is to maximize the rate of wood that can be harvested per unit of land and per unit of time.

8. Considering the full lifecycle of a single tree (plant, wait, harvest), what is the optimal time to cut down the entire tree to maximize the amount of wood harvested per unit of time? For this question, a numerical solution is adequate (eg using excel). In this case, use
r = 0.1, t0 = -2 years, k = 15m.



My answers are as follows:

1. 1.495 m
2. 0.001103 m
3. 1.77227 years
4. 0.57 years
5. k
6. r=ln((k/100)-1)/t0
7. Max growth rate occurs when t=0
8. 2


BTW, I think question 5 is incorrectly worded. It does NOT appear to depend on r. Am I correct?


I am pretty sure my answers to questions 1,2,3 and 5,6,7 are correct. I can share my methodology if requested.

I assume in order to answer question 4. the weight doubles as height doubles, weight doubles as width doubles and weight doubles as depth doubles, so

((1+0.495)^3n)=2, hence n = 0.57 years

My answer to question 8 is a guess.

Any feedback is greatly appreciated.
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Smaug123
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1 is correct. 2 is sort of right: the answer is 1.001103, and it has no unit - well, its unit is (metres per day) per (metres per day).

3 you have a typo: 1.7227, not 1.77227.
4 is correct, and as far as I can tell, so is 5.
6 is wrong. The height at time t=0 is \dfrac{k}{1+e^{-r t_0}}, which when divided by the maximal height (k) gives \dfrac{1}{1+e^{-r t_0}}.
7 should be correct - I don't see how it isn't 0.

I'll have a look at 8 and get back to you.
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Smaug123
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(Original post by smileatyourself)
Below are some questions on financial calculations and calculus for a job interview.

Alas I was not successful but have not been given any feedback from the company.

Maybe I answered some of the questions incorrectly, so I would like to ask readers here to help.



In the early life of a certain kind of tree, it is observed to grow at an exponential rate, whereby at the end of every day it is a fixed fraction taller than it was at the end of the previous day. It is observed that at 1 year it is 1m tall and at 5 years it is 5m tall.

1. How tall was the tree at 2 years?

2. What is the daily, fractional growth rate?

3. How long does it take to double in height?

As the tree grows, it keeps the same proportions. i.e. it grows in width and depth at the same rate as it grows in height.

4. How long does it take to double in weight?

When studied over a longer period of time, it is noticed that the fractional growth rate decreases as the tree gets older. It is suggested that the height h may follow the relationship h = k/(1+e^(-r(t+t0))) for some parameters k, r and t0 and where the tree is planted at time t=0.

5. Assuming this model, what is the maximum height of the tree, in terms of k and r?

6. Suppose a tree can grow to a maximum height of 100 times the height when it is planted. Find a formula for r in terms of t0.

7. At what point in time (again, in terms of k, r and t0) is the tree growing at its maximum rate?

In a managed forest, the goal is to maximize the rate of wood that can be harvested per unit of land and per unit of time.

8. Considering the full lifecycle of a single tree (plant, wait, harvest), what is the optimal time to cut down the entire tree to maximize the amount of wood harvested per unit of time? For this question, a numerical solution is adequate (eg using excel). In this case, use
r = 0.1, t0 = -2 years, k = 15m.
As far as I can tell, the answer to 8 is "immediately". After all, it's assumed we plant a tree of nonzero mass (in fact, the amount of wood harvested is roughly 6.75m^3, assuming trees are cubical), so we get infinite wood-per-unit-time.

However, assuming a different wording of the question ("maximise the gain of wood per unit time"), there is a definite answer, and it's roughly 19.9 years. Hint: define a "quantity" function q to be height^3. Then the amount of wood gained per unit time at time t is \dfrac{q(t)-q(0)}{t}.
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smileatyourself
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(Original post by Smaug123)
As far as I can tell, the answer to 8 is "immediately". After all, it's assumed we plant a tree of nonzero mass (in fact, the amount of wood harvested is roughly 6.75m^3, assuming trees are cubical), so we get infinite wood-per-unit-time.

However, assuming a different wording of the question ("maximise the gain of wood per unit time"), there is a definite answer, and it's roughly 19.9 years. Hint: define a "quantity" function q to be height^3. Then the amount of wood gained per unit time at time t is \dfrac{q(t)-q(0)}{t}.
Thanks for your help.

So at least I know I've got something in the region of about 6/8 correct.
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Smaug123
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(Original post by smileatyourself)
Thanks for your help.

So at least I know I've got something in the region of about 6/8 correct.
Yep, it looks like it.
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slut736
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(Original post by Smaug123)
Yep, it looks like it.
Hello, would you mind briefly sharing how you went about solving the questions? Thanks!
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Smaug123
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(Original post by slut736)
Hello, would you mind briefly sharing how you went about solving the questions? Thanks!
Q6, I gave the formula for the ratio of the height at time 0 to the maximal height (at time infinity) already.

Q8, I gave the "quantity" function. You just need to maximise the wood-per-unit-time function I also gave.
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