arboretum in psu

Question

Let L = f(h) be the amount of water (in liters) a species of tree needs per week when it is h meters tall. A particular tree of this species in the PSU arboretum is h meters tall, and so it requires f(h) liters of water per day, and we define v = f(h). Assume that f is invertible.

(a) What is the practical meaning of the expression f(h + 3)?

(b) What is the practical meaning of the statement f(h + 1) = v + 3?

(c) What is the practical meaning of f(h/2)?

(d) What is the practical meaning of f(h − 5) = v/2?We now let g(t) be the height of another tree of this species t years after 2009. For these parts, give your answer in terms of the functions f, g, f −1 , and/or g −1 .

(e) How much water did this tree need in 2013?

(f) How tall is a tree that needs 55 L per week?

(g) In what year did this tree need 40 L per week?

(h) When was this tree 9 m tall?

Answer 1

Hi Baris B.,

This problem requires applying the various functions to determine height from age, and water from height. In math terms, what you are applying are "convoluted functions": the output of one function is the input of the next function.

So the initail questions are just asking you to give an rephrasing in English of what the expressions stand for.

(a) f(h+3) is how much water the tree requires when it's 3m higher than h

(b) f(h+1) = v+3 indicates that for each meter taller, the tree requires 3L/day more

(c) f(h/2) = how much water the tree requires if it's only half as tall as h

(d) f(h-5) = v/2 indicates that the tree would require only half as much water if it were 5m shorter. Note that this expression contradicts the statement in (b)!

(e) in 2013, the tree was g(4) m tall, hence it required f(g(4)) L per day

(f) needs 55L = f^-1(55/7) m (note the conversion from per week to per day)

(g) 40L/week = f^-1(40/7) m tall, so this took t = g^-1(f^-1(40/7)) years old, and the year was 2009 + t

(h) was 9m tall when g(age) = 9m, so age = g^-1(9) . This was in year 2009 + age.

This problem actually pulls a "fast one" on you. You are given the two functions, allegedly for different specific trees, but the problem questions imply that you should consider both equations as applying to the tree(s) in the problem, since otherwise you couldn't solve through.

-- Cheers, -- Mr. d.