I ran into an interesting trigonometry problem a few weeks ago, when I was working on the cable pole installation. The problem was simple enough: extend the pole such that the trash trucks wouldn’t snag it, thus ripping it down. But the real question was: how tall must I extend it to get adequate clearance? Indeed.

Let’s take a look at the problem ^{1} a little closer. As I tend to be a visual person, I have to start with a drawing:

This is a crude diagram of my problem. The trapezoidal shape represents my aluminum pole and cable line and its geometric relationship to the alley and road. To help with all these visuals, I’ll show you the alley again (click for a bigger view):

As with all math problems, one should start with the knowns. I analyzed the pole and its surrounds by taking some measurements. Here’s what I know:

- pole height = 8.5ft
- distance from the pole base to the alley road = 6ft
- width of the alley road = 13ft
- at the current pole height, the height of the cable from the start of the alley = 11ft

The goal in any mathematical problem is to find as many equations describing the system as variables. This will allow you to solve the problem. So, let’s solve it! ^{2}

The next step would be to define our variables:

The blue colored vertical line on the far left of my diagram above is the original height of the pole. The red line above it, which I will call [pmath]x[/pmath], is the unknown extension of the pole. The blue colored vertical line on the far right labeled [pmath]z_pole[/pmath] is the height of the electrical pole across the alley. Thus, the angled blue line is the original pitch of the cable as it stretched from the electrical pole to my aluminum pole.

The angle that blue cable makes with the horizontal is [pmath]theta_1[/pmath], while the higher pole height makes angle [pmath]theta_2[/pmath]. I call the rise of the old pole height to be [pmath]z'[/pmath], while the rise of the electrical pole from the aluminum pole to be [pmath]z[/pmath]. I’ve defined [pmath]y[/pmath] as the rise of this new hypotenuse and [pmath]z_2[/pmath] to be the extended rise over the electrical pole.

Now we can start to define some trigonometric identities. I’ll be using the tangent based on our above knowns. First, we can state the old pole height as:

[pmath]tan theta_1=z/(6+13)=z^1/6[/pmath]

While we know that:

[pmath]z^1=11-8.5[/pmath]

We have our first variable solution:

[pmath]z^1=2.5[/pmath]ft

Therefore:

[pmath]z=7.9[/pmath]ft and [pmath]z_pole=16.4[/pmath]ft

Now, we also know that:

[pmath]tan theta_2=y/6=z_2/(6+13)[/pmath]

Let’s clean it up:

**Equation #1: ** [pmath]y=6/19*z_2[/pmath]

We can express [pmath]z_2[/pmath] in terms of [pmath]z_pole[/pmath]:

[pmath]z_2=16.4-(8.5+x)[/pmath]

Simplified:

**Equation #2: ** [pmath]z_2=7.9-x[/pmath]

Substituting Equation #2 into #1 and we have:

**Equation #3: **[pmath]y=6/19*(7.9-x)[/pmath]

Now the desired clearance will be:

[pmath]h_clearance=8.5+x+y[/pmath]

Let’s set that to be necessarily 14′. ^{3}. So now we have another relation:

**Equation #4: ** [pmath]x=5.5-y[/pmath]

There we have it, two equations left, two unknowns. Let’s substitute Equation #4 into #3 and we have:

[pmath]y=1.32[/pmath]ft and the most important:

[pmath]x=4.18[/pmath]ft

So I bought a 4 foot extension pole and got about 14 feet of cable clearance. A little hard math meets the real world.

#### Footnotes

- It occurred to me only after the project was initially complete that I might not have enough clearance for passing trucks in the alley. That’s what I get when I don’t put in enough time on the design side and speed ahead into production!
- And I must thank the wonderful PHP Math Publisher plugin!
- I called the local trash company to find out how big these trucks are