The distance d is 18 feet.
The height of the tower is 24 feet.
Explanation:
The length of the wire is 30 feet.
The height of the tower is ![(6+d)](https://tex.z-dn.net/?f=%286%2Bd%29)
where d is the distance of the towers base to the end of the wire.
Using Pythagorean theorem, we have,
![(6+d)^{2}+d^{2}=30^{2}](https://tex.z-dn.net/?f=%286%2Bd%29%5E%7B2%7D%2Bd%5E%7B2%7D%3D30%5E%7B2%7D)
Simplifying, we have,
![36+12 d+2 d^{2}=900](https://tex.z-dn.net/?f=36%2B12%20d%2B2%20d%5E%7B2%7D%3D900)
![2 d^{2}+12 d-864=0](https://tex.z-dn.net/?f=2%20d%5E%7B2%7D%2B12%20d-864%3D0)
Using quadratic formula, we shall determine the value of d.
![d=\frac{-12 \pm \sqrt{12^{2}-4 \cdot 2(-864)}}{2 \cdot 2}](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B-12%20%5Cpm%20%5Csqrt%7B12%5E%7B2%7D-4%20%5Ccdot%202%28-864%29%7D%7D%7B2%20%5Ccdot%202%7D)
![d=\frac{-12 \pm \sqrt{144+6912}}{4}](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B-12%20%5Cpm%20%5Csqrt%7B144%2B6912%7D%7D%7B4%7D)
![d=\frac{-12 \pm \sqrt{7056}}{4}](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B-12%20%5Cpm%20%5Csqrt%7B7056%7D%7D%7B4%7D)
![d=\frac{-12 \pm 84}{4}](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B-12%20%5Cpm%2084%7D%7B4%7D)
Thus, we have,
and ![d=\frac{-12 - 84}{4}\\d=\frac{-96}{4} \\d=-24](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B-12%20-%2084%7D%7B4%7D%5C%5Cd%3D%5Cfrac%7B-96%7D%7B4%7D%20%5C%5Cd%3D-24)
Since, the value of d cannot be negative.
Thus, ![d=18](https://tex.z-dn.net/?f=d%3D18)
Hence, the distance from the tower's base to the end of the wire is 18 feet.
Substituting
in
, we get,
![6+18=24](https://tex.z-dn.net/?f=6%2B18%3D24)
Thus, the height of the tower is 24 feet.