Mathematical models

P

Probability

Full Member
Joined
Jan 26, 2012
Messages
431
I have a tree trunk and I want to estimate the volume. I measure the length and find it to be 1.5 m. I put a tape measure round the trunk and calculate the circumference to be 92 cm.

Using the formula;

V = L x D^2 / 4 x pie

I calculate the volume; hence

V = 1.5 x 0.92^2 / 4 / pie

V = 0.10 m^3 (to 2 s.f.)

I'm then asked to carryout the same calculation using a calculator. In this example I'm using a CASIO fx-85GT PLUS

I'm told that the most obvious way of calculating the above using a calculator is to use the fraction function. An alternative way is to use the division key.

using the fraction function;

1.5 x 0.92^2 / 4 / pie = 0.997... (This example is exactly has I typed it in the calculator)

Now I tried this method;

1.5 x (92/100)^2 / 4 / pie = 0.997... (this example uses parenthesis)

1.5 x (92/100^2) / 4 / pie = 1.098... (this example squared everything in the parenthesis)

This method was even worse;

1.5 x (92^2/100) / 4 / pie = 10.103... (this example I squared the 92)

I can't seem to get the correct answer using the fraction function of a calculator. The correct answer is 0.10 (to 2 s.f.)

Now if I use the division function on the calculator I get;

1.5 x (92/100)^2 / 4 / pie = 0.101.....

Am I missing a technique using the calculator or is this type of problem not able to be solved using the fraction function of calculators?
 
First of all, I would avoid this: a/b/c. It's ambiguous. Are you dividing a by b, then by c? Or b by c first and then a by the result?
 
I'm using this / as a division line. hence;

1.5 x 0.92^2 / 4 / pie =

This means 1.5 times 0.92 and then divided by 4 which is then divided again by pie.

I have a Latex booklet somewhere but at the moment unable to find it which provides details how to post using it. Something else I need to learn.
 
Probability said:
This method was even worse;

1.5 x (92^2/100) / 4 / pie = 10.103... (this example I squared the 92)
Click to expand...
I don't understand. If you are squaring different terms, why are you surprised that the answer is different???

Regarding the initial formula, could you explain how circumference became diameter? And xPI became /PI? (And see my post about a/b/c)
 
Yes I understand it comes across confusing. In the booklet it is written as 4 x pie (using the pie symbol). I was always told that once a division has been carried out and further calculations under the division line must also be calculated by division, hence;

1.5 x 0.92 = If you enter this into a calculator and multiply under the division line instead of as written the answer will be incorrect.
4 / pie

Have I been misinformed about this!

Looking now again 1.5 x 0.92^2 / 4 x pie = 0.101... (This is correct)

I was told by a mathematician years ago that after doing a division that further calculations under the division line must also be carried out by further division. (Now I'm not so sure he was correct).

I'm not surprised about the answers being different except that trying to follow the instructions I'm not told that the answers would or should be different. I'd of expected them to be the same, whether hand written or calculated using calculators.

Regarding the formula, english wording I've used as an explanation. I could have said I measured the diameter directly, but actually mathematically you can't, hence further work would be required using another formula to work out the diameter from the circumference of the circle.

Bold is edited text
 
Last edited:
Probability said:
Yes I understand it comes across confusing. In the booklet it is written as 4 x pie (using the pie symbol).
Click to expand...
Could you post a photo of this formula?

I was always told that once a division has been carried out and further calculations under the division line must also be calculated by division, hence;

1.5 x 0.92 = If you enter this into a calculator and multiply under the division line instead of as written the answer will be incorrect.
4 / pie
Click to expand...

V = L x D^2 / 4 x π is not a clear way to write it.
How about this: V = (L x D^2 / 4) x π
Now we know that π is in the numerator (we multiply the result by it, not divide).
Yes, 4/π is correct if the whole thing is written as a fraction, since the order of operations is now clearer.


Have I been misinformed about this!
Click to expand...
Sorry, are you asking or making a statement?


I was told by a mathematician years ago that after doing a division that further calculations under the division line must also be carried out by further division. (Now I'm not so sure he was correct).
Click to expand...

Don't understand this rule.

I'm not surprised about the answers being different except that trying to follow the instructions I'm not told that the answers would or should be different. I'd of expected them to be the same, whether hand written or calculated using calculators.
Click to expand...
Which instructions tell you to square 92 only?

Regarding the formula, english wording I've used as an explanation. I could have said I measured the diameter directly, but actually mathematically you can't, hence further work would be required using another formula to work out the diameter from the circumference of the circle.
Click to expand...

If you want we can go over how to calculate the diameter based on the circumference. But if you don't know how to do it you should not just skip this step and simply continue using the number is the diameter. This is confusing to readers and will probably confuse you if you go back to read this later.
 
lev888 said:
Could you post a photo of this formula?

YES

V = L x D^2 / 4 x π is not a clear way to write it.
How about this: V = (L x D^2 / 4) x π
Now we know that π is in the numerator (we multiply the result by it, not divide).
Yes, 4/π is correct if the whole thing is written as a fraction, since the order of operations is now clearer.

NO π is not the numerator in this formula.

Sorry, are you asking or making a statement?

I think I am asking. I'm not sure however if different rules apply to different branches of mathematics!


Don't understand this rule.

Engineering and other mathematical sciences could have different rules!

Which instructions tell you to square 92 only?

Look at the formula I'm posting and see D^2

If you want we can go over how to calculate the diameter based on the circumference. But if you don't know how to do it you should not just skip this step and simply continue using the number is the diameter. This is confusing to readers and will probably confuse you if you go back to read this later.
Click to expand...

We don't need to go over how to calculate the diameter. The diameter is d = C / pie. This formula is slightly different in that it is not using radius squared but using the distance D round the log. This is where I said I was measuring around the circumference of the log. I understand all this becomes confusing quickly when variables keep changing like this. formula for volume of log.jpg
 
I think what is happening here is that the calculator is following the rules of BIDMAS when working out fractions, using the fractions function, and by using the calculator manually with the division function, (and not the fractions function) then anything below the division line must be divided otherwise the answer is incorrect.

For a person learning this subject of math's, I still maintain that technology will kick a persons legs from under them until a person fully understands the math's written on paper. I truly believe this.
 
Your formula is [MATH]V = \frac{LD^2}{4\pi}[/MATH]. I have no idea why it is using D to mean circumference, but it is correct.

When you evaluate a formula written as a fraction like this, the best practice is to be aware that the fraction bar groups the numerator and denominator, so you should always (unless you are certain otherwise) use parentheses to explicitly indicate that grouping. So inline (whether on paper or in a calculator), you are evaluating [MATH](L\times D^2)\div(4\times\pi)[/MATH].

For efficiency, you can change that to [MATH]L\times D^2 \div 4\div\pi[/MATH], but that is not required, and should only be done when you understand why.

Now, when you replace a variable in a formula with an expression, you should likewise put that expression in parentheses to make sure it is evaluated first. If you replace [MATH]L[/MATH] with [MATH]1.5[/MATH] and [MATH]D[/MATH] with [MATH]92\div 100[/MATH], then the result is [MATH](1.5\times (92\div 100)^2)\div(4\times\pi)[/MATH]. You need to be squaring the expression [MATH]92\div 100[/MATH] as a whole.

Now, if your calculator is a type that takes input graphically, some of this would change, as you could actually use the fraction bar notation. I haven't looked up your particular model.
 
The tree is basically a cylinder. The area of the base, being a circle, is pi*r^2. We multiply by the 3rd dimension which is height, h. So the volume for the tree is V = pi*r^2*h. Your formula is correct but the one I quoted is the standard formula for the volume of a cylinder.

The circumference is C = 2pi*r =92 cm. So r = 46/pi

V = pi*(46cm/pi)^2*(1.5m)=.....
 
Jomo said:
The tree is basically a cylinder. The area of the base, being a circle, is pi*r^2. We multiply by the 3rd dimension which is height, h. So the volume for the tree is V = pi*r^2*h. Your formula is correct but the one I quoted is the standard formula for the volume of a cylinder.

The circumference is C = 2pi*r =92 cm. So r = 46/pi

V = pi*(46cm/pi)^2*(1.5m)=..... 0.101...
Click to expand...
Thank you for your reply. While it is possible to use different formulas than the one provided in the coursework book, there could well be an underlying reason for the chosen formula that might be used again later in the material, hence it might not therefore be good to introduce another formula which might then lead to confusion further into the coursework material. (might do) I don't know!
 
Not the basic point but

[MATH]\dfrac{a * b * c}{d * e * f}[/MATH] means that you divide the PRODUCT of a, b, and c by the PRODUCT of d, e, and f. The horizontal line (the technical name of which I have forgotten) is a grouping symbol for both numerator and denominator. I suspect that is what your mathematician friend told you. Technically, the slash does work the same way through PEDMAS if there are only slashes, but people make mistakes on that all the time. Consequently, no one who reads a/b/c will be confident whether you meant

[MATH]\dfrac{a}{\dfrac{b}{c}} = \dfrac{ac}{b}[/MATH] or

[MATH]\dfrac{\dfrac{a}{b}}{c} = \dfrac{a}{bc}.[/MATH]
And of course, any operator other than a slash destroys the whole game.

a/b+c/d IS NOT EQUAL TO

[MATH]\dfrac{a}{\dfrac{b + c}{d}} = \dfrac{ad}{b + c}.[/MATH]
a/b+c/d in fact equals

[MATH]\dfrac{ad + bc}{bd}.[/MATH]
Do not rely on PEDMAS when using the slash to indicate division. Any complexity in your expression when using slashes will introduce uncertainty that you can avoid by using explicit grouping symbols.
 
Last edited:
JeffM said:
Not the basic point but

[MATH]\dfrac{a * b * c}{d * e * f}[/MATH] means that you divide the PRODUCT of a, b, and c by the PRODUCT of d, e, and f. The horizontal line (the technical name of which I have forgotten) is a grouping symbol for both numerator and denominator. I suspect that is what your mathematician friend told you. Technically, the slash does work the same way through PEDMAS if there are only slashes, but people make mistakes on that all the time. Consequently, no one who reads a/b/c will be confident whether you meant

[MATH]\dfrac{a}{\dfrac{b}{c}} = \dfrac{ac}{b}[/MATH] or

[MATH]\dfrac{\dfrac{a}{b}}{c} = \dfrac{a}{bc}.[/MATH]
And of course, any operator other than a slash destroys the whole game.

a/b+c/d IS NOT EQUAL TO

[MATH]\dfrac{a}{\dfrac{b + c}{d}} = \dfrac{ad}{b + c}.[/MATH]
a/b+c/d in fact equals

[MATH]\dfrac{ad + bc}{bd}.[/MATH]
Do not rely on PEDMAS when using the slash to indicate division. You will introduce uncerttainty that you can avoid by using grouping symbols if there is any complexity.
Click to expand...
The horizontal line (the technical name of which I have forgotten)

Vinculum as I remember it.

Vinculum. A horizontal line placed over an expression to show that everything below the line is one group. ... In some countries it is the horizontal line used to separate the numerator and denominator in a fraction, also called a "fraction bar".
 
My classic example is [math]\dfrac {100}{ \frac {10}{2}}[/math] vs [math]\dfrac{\frac{100}{10}}{2}[/math]
Are these the same?
 
Jomo said:
My classic example is [math]\dfrac {100}{ \frac {10}{2}}[/math] vs [math]\dfrac{\frac{100}{10}}{2}[/math]
Are these the same?
Click to expand...
There are some rules in math that decide what the outcome is going to be. Some rules (just an example) are BIDMAS, some countries call that BODMAS and some have another name I can't recall at this time.

In your two examples, taking your first example, divide the lower numerator 10 by the denominator 2 and record the quotient and then divide 100 by the quotient to give 20.

In your second example, divide the numerator 100 by 10 to record the quotient and then divide the new numerator by 2 to equal 5

For me the giveaway is the length of the vinculum which tells me there is either a fraction below it or above it. It's a long time since I've seen fractions like those and the way I carried those out above is how I remember doing them many many moons ago!!
 
You must log in or register to reply here.
Top