Stuck with my year 12 Logarithmic and Exponential Functions Assignment

[kmark.64]

Hello everyone,

The question i'm stuck on is:

The following data refers to the distance to the horizon from various elevation points. The distance (D) is measured in kilometers and the height (H) is measured in meters above sea level.

H (Meters)	D (Km)
1	3.55
1.5	4.4
2	5
2.55	5.7
3	6.2
3.5	6.7
4	7.15

The visible distance to the horizon (D) depends on the what your elevation is. Model the data and find a mathematical rule for D as a a function of H.
Use your rule to predict what elevation is needed to be able to see 40kms to the horizon.

I was just wondering how I should begin, and/or what I should do?
Some people have said it's a power function of some sort but I'm unsure...
So if anyone could help that would be amazing

Thank you <3

 

[Deleted member 4993]

Hello everyone,

The question i'm stuck on is:

The following data refers to the distance to the horizon from various elevation points. The distance (D) is measured in kilometers and the height (H) is measured in meters above sea level.

H (Meters)	D (Km)
1	3.55
1.5	4.4
2	5
2.55	5.7
3	6.2
3.5	6.7
4	7.15

The visible distance to the horizon (D) depends on the what your elevation is. Model the data and find a mathematical rule for D as a a function of H.
Use your rule to predict what elevation is needed to be able to see 40kms to the horizon.

I was just wondering how I should begin, and/or what I should do?
Some people have said it's a power function of some sort but I'm unsure...
So if anyone could help that would be amazing

Thank you <3

Have you plotted the points? May be use a spreadsheet software like excel.

 

[stapel]

The following data refers to the distance to the horizon from various elevation points. The distance (D) is measured in kilometers and the height (H) is measured in meters above sea level....

The visible distance to the horizon (D) depends on the what your elevation is. Model the data and find a mathematical rule for D as a a function of H. Use your rule to predict what elevation is needed to be able to see 40kms to the horizon.

I was just wondering how I should begin, and/or what I should do?

What methods did they give you in class for this sort of exercise? What does your textbook show? There are various methods; you'll need to tell us the method(s) they told you to use, and how far you've gotten with this info. Thank you!

 

[kmark.64]

What methods did they give you in class for this sort of exercise? What does your textbook show? There are various methods; you'll need to tell us the method(s) they told you to use, and how far you've gotten with this info. Thank you!

The assignment is a complex non-routine piece of assessment, meaning we are not shown in class how to do it.
The unit in my textbook only shows the exponential growth and decay rate formula F(x) = ar^x.
The examples shown in the textbook don't help, as all the numbers used are of the same growth rate factor.
Plus I have plotted the points using Excel, there is quite an obvious trend but we are required to show our working of how we created the formula :/

I'm sorry I cant explain much else I just really don't understand. But if you could possibly suggest anyway I should go about it that would be awesome. :smile:

Thank you

 

[stapel]

The assignment is a complex non-routine piece of assessment, meaning we are not shown in class how to do it.
The unit in my textbook only shows the exponential growth and decay rate formula F(x) = ar^x.

Hm... Too bad you can't use F(x) = ax^b. That form gives a really nice match to the data.

The examples shown in the textbook don't help, as all the numbers used are of the same growth rate factor.

I'm not sure what you mean by this...? Are you saying that all the examples in the book used points which were exactly on the same curve? If so, then I'm guessing what you're supposed to be doing here is a regression in your graphing calculator (or in a spreadsheet, maybe).

Plus I have plotted the points using Excel, there is quite an obvious trend but we are required to show our working of how we created the formula

Investigate the math tools in your spreadsheet software. You may need first to convert your output values to logged values, because I think Excel only does linear regressions. Since logging exponential outputs will create a linear-ish transformed set of outputs, Excel can then provide a linear regression, which you can convert back to exponential. They show how this might work here.

Of course, I'm only guessing. I'm not in your classroom and I don't have your book. If they've really taught you nothing about this, then it's anybody's guess how they're expecting you to proceed. :shock:

 

[HallsofIvy]

A "power function" is a function of the form f(x)= ax^r, not an exponential or logarithmic function. Now, exactly what are you to do with this? Whether you use f(x)= ax^r, a "power function", or f(x)= ar^x, an exponential function, you have two parameters, a and r, to find. One method is to choose two points, preferably distant from one another, on the graph to get two equations to solve for a and r and get a function that will pass precisely through those two points and close to the others. The other much, much harder, is to seek a "root mean square" approximation that will minimize sum the square of the errors between computed point and given point for every point.

 

[JeffM]

The assignment is a complex non-routine piece of assessment, meaning we are not shown in class how to do it.
The unit in my textbook only shows the exponential growth and decay rate formula F(x) = ar^x.
The examples shown in the textbook don't help, as all the numbers used are of the same growth rate factor.
Plus I have plotted the points using Excel, there is quite an obvious trend but we are required to show our working of how we created the formula :/

I'm sorry I cant explain much else I just really don't understand. But if you could possibly suggest anyway I should go about it that would be awesome. :smile:

Thank you

Are you or are you not allowed to use excel or a hand calculator to do linear regression?

If not, have you been taught how to do linear regression with pencil and paper?

If so, you can turn exponential relationships into linear ones as follows:

\(\displaystyle y = ar^x \iff log(y) = log(a) + xlog(r).\)

Let z = log(y), b = log(a), and m = log(r).

\(\displaystyle log(y) = log(a) + xlog(r) \iff z = b + mx.\) The latter is linear in form.

Thus, an exponential relationship between y and x implies a linear relationship between the log of y and x. So regress between x and the log of y, and look at the coefficient of correlation.

Similarly, you can turn power relationships into linear ones as follows:

\(\displaystyle y = ax^r \iff log(y) = log(a) + rlog(x).\)

Let z = log(y), b = log(a), and u = log(x).

\(\displaystyle log(y) = log(a) + rlog(x) \iff z = b + ru.\) The latter is linear in form.

Thus, a power relationship between y and x implies a linear relationship between the log of y and the log of x. So regress between the log of x and the log of y, and look at the coefficient of correlation.

Remember that b = log(a) and that log(r) is not r. So you must adjust your final formulas in x and y accordingly.

There are some subtleties associated with using this technique that I forgot about long ago. If the two processes both give good correlation coefficients, I might see which gives a lower sum of the squared errors terms using the final non-log formulas and whether those error terms themselves appear correlated. Correlated error terms indicate a problem. But, as I implied, it has been a long time since I studied statistics.