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## #1 2021-12-06 05:10:28

Hannibal lecter
Member
Registered: 2016-02-11
Posts: 330

### percentage growth vs actual growth meaning

Hydroelectric power is electric power generated by the
force of moving water. Figure 1.15 shows6 the annual
percent growth in hydroelectric power consumption by
the US industrial sector between 2004 and 2009.

(a) Find the largest time interval over which the percentage growth in the US consumption of hydroelectric
power was a decreasing function of time. Interpret
what decreasing means, practically speaking, in this
case.

(b) Find the largest time interval over which the actual
US consumption of hydroelectric power was a decreasing function of time. Interpret what decreasing
means, practically speaking, in this case.

I know (a) answer is the percentage growth was a decreasing function from (2005 to 2007)
but in (b) the book solution said actual US consumption interval was (2004–2007)
what is the difference before I see the book solution I thought it's also (2005 to 2007) same answer!
it's being two days I'm thinking about that and I didn't get the key
how it's 2004 to 2007???!!! the consumption is increased from 2004 to 2005 then decreased from 2005 to 2007
so we shouldn't include 2004

Wisdom is a tree which grows in the heart and fruits on the tongue

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## #2 2021-12-06 22:55:31

Bob
Registered: 2010-06-20
Posts: 9,358

### Re: percentage growth vs actual growth meaning

The scale on the 'y' axis is "percent growth".  And it has negative values as well as positive.  So any points on the lines that are below the x axis means the 'growth' is negative ie. not growing but rather shrinking.

You are interpreting a line with a positive gradient (slope) as meaning a positive growth. This is where you are being misled.  A line that slopes upward means the growth is getting less negative and eventually the line crosses the x axis and so real positive growth begins.

This doesn't happen until a point between 2007 and 2008.  The exact moment is not necessarily in the last part of 2007. Single known data points are plotted and then the grapher has assumed that it is OK to join them up with straight line segments.  This may not be true.  The curve between the 2007 point and the 2008 point might actually be more like a quadratic curve which would put the crossing x axis point even later. Or the curve light shoots up rapidly at the start of 2007 and crosses the axis much sooner.  We just don't know.

Also we don't know the moment when the 2007 data is collected into a point on the graph.  You'd have to wait until the end of the year before you had all the data for that year so, thinking about it some more, the 2007 point is actually 31st December 2007.

In which case the growth is negative right up to the end of 2007 as the book says.

So, when you see a graph, think carefully about what it is telling you and also what it isn't telling you.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #3 2021-12-07 05:11:35

Hannibal lecter
Member
Registered: 2016-02-11
Posts: 330

### Re: percentage growth vs actual growth meaning

Bob wrote:

You'd have to wait until the end of the year before you had all the data for that year so, thinking about it some more, the 2007 point is actually 31st December 2007.

Bob

I understand you completely and you helped me a lot in that specially when you told me about negative and positive growth
but there is still a problem in the question please check that another problem with same answers and you will notice the conflict

Solar panels are arrays of photovoltaic cells that convert
solar radiation into electricity. The table shows the annual percent change in the US price per watt of a solar
panel :

a)Find the largest time interval over which the percentage growth in the US price per watt of a solar panel
was an increasing function of time. Interpret what
increasing means, practically speaking, in this case.

(b) Find the largest time interval over which the actual
price per watt of a solar panel was an increasing
function of time. Interpret what increasing means,
practically speaking, in this case.

please see the "a" branch is completely clear and the book answer is clear, the answer is from (2004-2006)
the problem again is in the actual graph, the book solution is : 2005-2006
you told me to chose the positive growth intervals right? look I draw it on notebook and you can see there is positive interval that the book ignored please see my pictures :-

there is positive interval I highlighted it in blue pen, please tell me what is happening is the book wrong or I'm not good enough to understand great ideas like this one
now because I'm confused I have to re-ask my question  what is the different between growth in branch "a" and growth in branch "b" I didn't see in webs something called " actual growth I searched a lot" do you have simple example in clear numbers because I understand most of such subjects in interactive way like (price in 2011 per watt for solar panel is 2\$ after growth 30% it become \$2.6... and so on I only understand it this way sometimes when things goes difficult when I see numbers become real things and represent real simple things in detailed I understand more,)
so I wish after you help me in this question and we done from it I wish next to make your own simple example to me with all detailed pls

Last edited by Hannibal lecter (2021-12-07 12:32:45)

Wisdom is a tree which grows in the heart and fruits on the tongue

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## #4 2021-12-07 22:38:17

Bob
Registered: 2010-06-20
Posts: 9,358

### Re: percentage growth vs actual growth meaning

hi Hannibal lecter

It's a confusing question; it took several readings to get the sense of it.  Then I used a spreadsheet with an additional row.

What I have done is made up a start figure for the price per watt before 2004.  I used 100.  You can use any figure and still get the answers you need.

The third row is the multiplier needed to calculate the new price after the % change is applied.  So for 2004 the price goes down by 5.7 % so the new price is (100-5.7)% = 94.3% If you multiply 100 by 0.943 you get the 2004 price.  And so on.

Check these figures and make sure you understand them.

Next I thought about when the % figure is calculated.  It cannot be applied until the end of the year so, where it says 2004 that is actually 31st December 2004.

So your graph has the same issue that the previous question had. By drawing a straight line between the points you are assuming the change is linear throughout the year.  This is very unlikely so assuming the line crosses the axis at a certain moment in the year is unlikely to be true.

Part (a) asks about the percentage growth being positive. This is 2005 to 2006. You have understood that ok.

Part (b) asks about the actual price.  That is where my extra row helps. Using my 'index' of 100 as the starting point you can see the price drops in 2004.  But after that the price recovers and is above 100 for the rest of the years.  So I think the answer to part (b) is 2005 to 2008.

Hope that helps,

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #5 2021-12-08 05:14:44

Hannibal lecter
Member
Registered: 2016-02-11
Posts: 330

### Re: percentage growth vs actual growth meaning

if I didn't understand this exercise 100% is that will effect on studying the next topic which is exponential functions
if I understand this 85% can I cross and go to another topic? the next topic will talk about exponential functions
can I start reading it or I have to complete these exercises and understand it 100% from all ways?

Last edited by Hannibal lecter (2021-12-08 05:18:58)

Wisdom is a tree which grows in the heart and fruits on the tongue

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## #6 2021-12-08 11:20:15

Hannibal lecter
Member
Registered: 2016-02-11
Posts: 330

### Re: percentage growth vs actual growth meaning

Bob wrote:

hi Hannibal lecter

Part (a) asks about the percentage growth being positive. This is 2005 to 2006. You have understood that ok.

Part (b) asks about the actual price.  That is where my extra row helps. Using my 'index' of 100 as the starting point you can see the price drops in 2004.  But after that the price recovers and is above 100 for the rest of the years.  So I think the answer to part (b) is 2005 to 2008.

Hope that helps,

Bob

you worked it very accurate and neat I like they way you work
do you think the book needed only intervals for each consecutive years because that it's ignored some values?
pls read the book full manual solution :

(a) The largest time interval was 2004–2006 since the percentage growth rate increased from −5.7 in 2004 to 9.7 in
2006. This means that from 2004 to 2006 the US price per watt of a solar panel grew relatively more with each
successive year.
(b) The largest time interval was 2005–2006 since the percentage growth rates were positive for each of these two
consecutive years. This means that the US price per watt of a solar panel steadily increased during the two year span
from 2005 to 2006, then decreased in 2007.

do you notice about "consecutive years"? I think the BookManual solution provider understand what the textbook wanted and he give it to him right?
because from what I see now both you and other answer looks acceptable expect your answer is more accurate

Wisdom is a tree which grows in the heart and fruits on the tongue

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## #7 2021-12-08 21:12:57

Bob
Registered: 2010-06-20
Posts: 9,358

### Re: percentage growth vs actual growth meaning

Is it time to move on?  I think it is.

Some people think you must learn math in a linear way; fully understand A before you move on to B.  I don't agree.  I think it's more like doing a jigsaw puzzle.  When you do a jigsaw you do the easy bits first and come back to the harder bits when there are less bits to work with.  There's nothing wrong with having areas still to master.  I'm still working on "contour integration" and hoping to use this to solve a problem I met nearly ten years ago.

If I'm reading a math book and get stuck, let's say on chapter 3, I go on to look at chapter 4, because I sometimes find that ch.3 becomes more understandable once I know where it's leading.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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