Building a Picnic Table

Someone writes

Jonathan and/or [redacted],

I’ve got a basic geometry question for either of you, that I need a little help with…

I am working on making a picnic table with [redacted] and we wanted to design our own legs for the table which are angled and cross. I have been able to work out all relationships between heights, lengths, widths, angles. etc. and now have two equations with two unknowns that will allow me to solve for the correct angle at which to cut the wood for the legs. Unfortunately I have forgotten how to solve basic cos, sin, and tan when variables are involved. The two equations are (all units are in inches, not that it matters):


$\displaystyle \textrm{sin} (\alpha)$ $\textstyle =$ $\displaystyle \frac{5.5}{x}$ (1)
$\displaystyle \textrm{tan} (\alpha)$ $\textstyle =$ $\displaystyle \frac{29}{35.5-x}$ (2)

How do I solve for $x$ and $\alpha$? Also, what is the answer :-)

Thanks,
[redacted]

Because

\begin{eqnarray*}\textrm{tan} (\alpha) & = & \frac{\textrm{sin} (\alpha)}{\text......rac{\textrm{sin} (\alpha)}{\sqrt{1 - \textrm{sin}^{2} (\alpha)}}\end{eqnarray*}

we have

$\displaystyle \frac{29}{35.5-x}$ $\textstyle =$ $\displaystyle \frac{\frac{5.5}{x}}{\sqrt{1 - \left(\frac{5.5}{x}\right)^{2}}}$ (3)

Simplifying and solving by computer algebra, we get

\begin{eqnarray*}\frac{3243}{841} x^2 + \frac{8591}{841} x - \frac{1017005}{3364} & = & 0\end{eqnarray*}

and

\begin{eqnarray*}x & = & \frac{ \pm 1276\sqrt{2071} - 8591}{6486}\end{eqnarray*}

Only the positive choice is a root of Eqns. (1) and (2), giving

\begin{eqnarray*}x & = &7.62835577107986... \\\alpha & = & 0.805235953662952... \\& = & 46.1366216570791... \textrm{(degrees)}\end{eqnarray*}

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2 Comments »

  1. Matt said

    Thanks for the solution Jonathan! How does the computer algebra program work, does it have a friendly interface and could I run it on my PC?

    Also we got your post card last week, I guess as long as we can visit Jeff and Hurk there is no need to go to Paris.

  2. wellons said

    I use Sage, which doesn’t do exactly what I want it to. I have to manually do some of the simplification of Eqn. (3) because otherwise it gets inexplicably hung up on things like “sqrt(x^2)”. I bring the denominators up by multiplying by them, then I move both terms to the same side of the equation. Here’s a sequence of commands that does the computer algebra simplification and solving in this blog entry:

    sage: expand((2 * x)^2 * (1 – (11 / (2 * x))^2) – (71 /2 – x)^2 * (11 / (29))^2 )
    3243*x^2/841 + 8591*x/841 – 1017005/3364
    sage: solve([ 3243*x^2/841 + 8591*x/841 – 1017005/3364 ], x)
    [x == (-1276*sqrt(2071) – 8591)/6486, x == (1276*sqrt(2071) – 8591)/6486]

    As you can see, Sage has a command line interface. The text following “sage: ” is text I typed. The other lines are results. Best of all, sage is free and seems to be a pretty active project. It looks like it has a windows version here: http://www.sagemath.org/download.html .

    There are more powerful computer algebra systems that cost money and are often purchased by labs and universities for their researchers. I would say to ask around.

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