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During its first year a species of tree reaches a height of 7cm. It continues to grow for another 9 years attaining its maximum height at 10 years of age. Each year it continues to grow at the rate of double the previous years height. At what age will the tree be half its maximum height?

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During its first year a species of tree reaches a height of 7cm. It continues to grow for another 9 years attaining its maximum height at 10 years of age. Each year it continues to grow at the rate of double the previous years height. At what age will the tree be half its maximum height?

@ 9 yrs?

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9.25 years

Every year the tree is 3 times as high as it was the previous year.

That means if it is x high, it will gain 2x throughout the year, totaling 3x at the end. We want to find when it is 1.5x

if it gains 2x every year, it will gain x every six months, or .5x every 3 months. Also stated as .5x every quarter of a year or .25 years.

So .25 through the year it will be half the height it will be at the end of the year.

Edited by Noct

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at 9 years it will be half the max if it doubles every year and 10 years is the max.

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The answers of 9 years would be correct if it doubled its height each year. But it doesn't. The rate at which it grows is double it's height. So if it is 5 feet high, it will grow 10 feet a year. So in one year it would be 15 feet, or three times it's height.

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bonanova    76
During its first year a species of tree reaches a height of 7cm. It continues to grow for another 9 years attaining its maximum height at 10 years of age. Each year it continues to grow at the rate of double the previous years height. At what age will the tree be half its maximum height?

Specifying a growth rate [distance/time] = 2 x height [distance] makes the problem unsolvable.

You'd need to say what units of time and height make the growth formula correct; the OP does not do this.

Assuming sloppy wording was used to say the tree doubles its height every year, [the only reasonable alternative] the answer is 9 years.

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Guest
Specifying a growth rate [distance/time] = 2 x height [distance] makes the problem unsolvable.

You'd need to say what units of time and height make the growth formula correct; the OP does not do this.

Assuming sloppy wording was used to say the tree doubles its height every year, [the only reasonable alternative] the answer is 9 years.

It is not the only reasonable alternative.

How i understood it, and what i contend is the most reasonable interpretation, is that each year it grows double it's height at the beginning of the year. Therefore it would triple its height each year.

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bonanova    76
It is not the only reasonable alternative.

How i understood it, and what i contend is the most reasonable interpretation, is that each year it grows double it's height at the beginning of the year. Therefore it would triple its height each year.

Hi Noct,

I was commenting, rather, about the OP:

First, I would say it's not a well posed problem.

Consider the significant differences among these three statements.

[1] Each year it continues to grow at the rate of double the previous years height.

[2] Each year its height changes by a multiplicative factor equal in magnitude to two times the current height of the tree expressed in cm.

[3] Each year its height increases by an amount equal to two times its current height.

Statement [1] is from the OP and is not well posed.

Statement [2] is well posed, and would be, to my thinking, what the OP tried to say.

If you take the OP as a formula for calculating next year's growth rate from this year's height:

[1] leaves you in the dark as to what units to use: cm, inches, feet, meters and furlongs all give different results.

[2] works, but isn't what the OP says.

[3] tells you how the height changes, but it isn't what the OP says; it doesn't address growth rate at all - rather a growth amount.

Second, a growth rate most commonly refers to the derivative of height with respect to time, and it has the units of [length]/[time].

If [2] is what is meant, calling a factor a rate is incorrect; they have different dimensions. Rate is dimensionless.

If [3] is what is meant, calling a height increase a rate is also incorrect. Summarizing,

Rate = [length/time]; height and height increase = [length] and factor = [dimensionless].

To say Rate = 2 x Height creates a dimensionality error.

To say Factor = 2 x Height also creates a dimensionality error.

To say Height increase = 2 x Height is dimensionally correct, but the OP purports to specify a rate.

So to my mind the OP does not say clearly how the tree grows.

Regarding my comment about alternative, I'm simply saying that if you abandon the idea of calculating a growth rate [or factor or height increase] from the tree's current height, you're pretty much left with taking the OP to say that the height, itself, doubles each year.

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Guest

Nice Q. It reminded me of a problem I got in my 9th grade 22 years ago, by my teacher to check if I was paying any attention. wow I am old.

What is product of :

(x - a) (x - b) ........ (x - z) ?

The question itself has a loop hole. If we pay attention to this question, it doubles every year and grows for 10 years, then when is it 1/2 the height. Well throw people off, we could have mentioned 22 years and still would have come up with the answer (22 - 1 ) years.

Nice brain teaser.

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Guest

YOU TRIED TO POSE A TRICK PUZZLE>>>WHICH TURNED UP INTO A ......

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bonanova    76
What is product of :

(x - a) (x - b) ........ (x - z) ?

I've seen this product expanded and I suppose I could search for it. Too lazy.

But clearly it has a x26 term and a abcdefghijklmnopqrstuvwxyz term.

What the other terms must look like can be seen by expanding the first four terms:

(x - a) (x - b)(x - c) (x - d) =

x4 [1] +

x3 [-a -b -c -d] +

x2 [ab + ac + ad + bc + bd + cd] +

x1 [-abc -abd -acd -bcd] +

x0 [abcd]

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There is no ambiguity in the question initially posted, only ,as pointed out, sloppy wording. The question was to imply "each year the tree attained a height of double the previous year" without being too obvious, and still allow an element of ambiguity. However the posted version "rate of growth" was too forthright and to this version Noct was correct, although not the answer I intended. It appears some of the answers are confusing themselves by mixing what I might have been implying (which would have been the correct train of thought ,had I worded differently) with what I actually posted. So right answer, wrong question.

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the tree would be half of its maximum age in its second last year coz it grows double every year

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