# Two posts, one 10 ft tall and other 20 ft tall, are set 30 feet apart from one another. A guy wire, used to stabilize the posts, is staked to the ground at some point between them. Express the...

Two posts, one 10 ft tall and other 20 ft tall, are set 30 feet apart from one another. A guy wire, used to stabilize the posts, is staked to the ground at some point between them. Express the length of the wire as a function of x, the distance between the stake and the shorter post.

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### 1 Answer

Please refer to the attached image. It is the figure that represents the given conditions in the problem.

Notice that when the wire was attached from the stake to each end of the poles,two right triangles are formed. So, to determine the length of each wire, apply the Pythagorean formula.

`c^2=a^2+b^2`

Using that, the length of the wire connected to the shorter post will be:

`(w_1)^2=x^2+10^2`

`(w_1)^2=x^2+100`

`w_1=sqrt(x^2 +100)`

And the length of the wire connected to the other post will be:

`(w_2)^2=(30-x)^2+20^2`

`(w_2)^2=900-60x+x^2+400`

`(w_2)^2=x^2-60x+1300`

`w_2=sqrt(x^2-60x+1300)`

Then, add the lengths of the two wires to get the total length.

`W=w_1+w_2`

`W=sqrt(x^2+100)+sqrt(x^2-60x+1300)`

**Thus, the length of the wires used (express as a function of x) is**

**`W(x)=sqrt(x^2+100)+sqrt(x^2-60x+1300).` **