Front Porch Math
By Bobby Neal Winters
The following is a real problem that a carpenter with a great respect for trigonometry came to me with. Say you are going to add a porch of width L to the side of a house and want to over it by adding to the roof as depicted in the picture below:
I must confess that this problem took longer for me to solve than it might have because I was guided initially by the carpenters love of trigonometry. It can be solved that way, but this is a nice case study of how imposing a coordinate system can simplify the problem. Consider the following:
The equation for the line representing the porch roof is y=mx and for the old roof is y=nx-nL. It is an exercise to work out that x*= nL/(n-m) and y*=mnL/(n-m). We may use the Pythagorean Theorem (or the distance formula) to calculate the value of D2=(x*)2+(y*)2 which after substitutions simplifies to:
D2=(1+m2)n2L2/(n-m)2 or D=(1+m2)1/2 nL/(n-m).
For the given values of n and m, this works out to D=2.02598L.
Having done it this way, I also worked it by trigonometry, but I despair of reproducing those calculations without specialized mathematical software.