0
$\begingroup$

Peace to all. I am taking a Physics course that is algebra-based and within one of our assignments, we have insert numbers into Excel and generate a number of graphs. After graphing said data (which was the easy part), we have to answer a series of questions about said graphs. One of the questions asks "how could this coefficient be written in terms of ฯ€?" I must admit I have no idea of how to do so. I am not asking for the answer but assistance on how one could obtain it. The Coefficient of the slope is "0.7854m".

I was given this hint "Rewrite the area formula ๐ด๐ถ=๐œ‹๐‘Ÿ2 in terms of the diameter, ๐‘‘=๐‘Ÿ/2. Rename ๐‘‘2 as ๐‘ฅ. Rename ๐ด๐ถ as y." It did not make sense to me though.

Edit: In Excel we have/had to input our own numbers labeling in the following columns "Diameter (m), Diameter^2/D^2 (m^2), Radius (m), Area (m^2)". For this particular question, I had to use data from the "Area (y-axis) Vs D^2 (x-axis)".

Area #'s are: 0.785398163, 1.767145868, 3.141592654, 4.908738521, 7.068583471, 9.621127502

D^2 #'s are 1 2.25, 4, 6.25, 9, 12.25

....after graphing I got "y=0.7854x" and r^2=1.

Area vs D^2

$\endgroup$
4
  • 1
    $\begingroup$ Numerically that's close to $\,\pi/4\,$, but the hint can't be understood unless you tell more about the context, and what "this coefficient" is. $\endgroup$
    – dxiv
    Aug 29, 2021 at 4:27
  • $\begingroup$ @dxiv Peace, I added more information in the question. $\endgroup$
    – יהודה
    Aug 29, 2021 at 4:47
  • 1
    $\begingroup$ "Area (y-axis) Vs D^2 (x-axis)" $\;-\;$ Assuming that means $A/d^2$ then the ratio is indeed $\,\pi/4\,$ since $A=\pi r^2=\pi d^2 / 4\,$. $\endgroup$
    – dxiv
    Aug 29, 2021 at 5:06
  • $\begingroup$ @dxiv I added the graph, for visuals $\endgroup$
    – יהודה
    Aug 29, 2021 at 5:12

2 Answers 2

1
$\begingroup$

I'm assuming your graph plots the area of a circle vs the diamater squared. If this is the case, then every $x$ is interpreted as a diameter squared, so we get $$ x = d^2 $$ On the other hand, since the $y$ value is interpreted as the area of a circle, then we get $$ y =\text{Area} = \pi r^2 $$ Now, since you numerically obtained the line equation $ y=0.7854 x$, this means that the slope is just the fraction $\frac{y}{x}$, or in other words $$ \frac{y}{x} = 0.7854 \tag{1} $$ The question you ask yourself now is, can this slope $\frac{y}{x}$ be written in terms of $\pi$? The answer is yes! We can use the formulas from the beginning to get this: \begin{align} \frac{y}{x}= \frac{\pi r^2}{d^2} \overset{\color{purple}{d=2r}}{=}\frac{\pi r^2}{(\color{purple}{2r})^2}= \frac{\pi r^2}{4r^2}= \frac{\pi}{4} \tag{2} \end{align} So combining equations $(1)$ and $(2)$ you get that your slope can be written in terms of $\pi$ as $$ 0.7854 \approx \frac{\pi}{4} $$ giving the desired interpretation, which can be verified with a calculator.

$\endgroup$
2
  • $\begingroup$ Thank you so much. What topic(s) or areas should I study to learn how to do this? $\endgroup$
    – יהודה
    Aug 29, 2021 at 5:34
  • 2
    $\begingroup$ The key thing here is not to forget what your variables $x$ and $y$ mean. Notice that the answer just pops out once you substitute the formula for area and the diameter squared, but to do this you need to remember what's the interpretation of your data. Don't get lost in the numbers, always remember what you're working with. $\endgroup$
    – Robert Lee
    Aug 29, 2021 at 5:40
1
$\begingroup$

The area of the circle is $\,A=\pi r^2\,$, or $\,A=\pi\left(\dfrac{d}{2}\right)^2=\dfrac{\pi}{4}d^2\,$ in terms of its diameter $\,d=2r\,$.

With $\,y=A\,$ and $\,x=d^2\,$ the graph of the area against the square of the diameter is $\,y=\dfrac{\pi}{4}x\,$.

  • The equation represents a straight line, so it makes sense to speak of the slope of the graph (otherwise, for a non-linear graph, the slope would change between different points).

  • The slope is $\,\dfrac{dy}{dx} = \dfrac{\pi}{4}\approx 0.7854\,$. Note that the slope has dimension $\,\dfrac{m^2}{m^2}=1\,$ i.e. it is dimensionless, so it is wrong to say that the slope is $\,0.7854 \color{red}{\,\text{m}}\,$.

$\endgroup$
1
  • $\begingroup$ Thank you for your help. It's weird because I understand but at the same time I didn't. Like my mind had to download a program to make it make sense. $\endgroup$
    – יהודה
    Aug 30, 2021 at 17:44

Your Answer

By clicking โ€œPost Your Answerโ€, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.