Answer :
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 [tex](6+d)[/tex]
where d is the distance of the towers base to the end of the wire.
Using Pythagorean theorem, we have,
[tex](6+d)^{2}+d^{2}=30^{2}[/tex]
Simplifying, we have,
[tex]36+12 d+2 d^{2}=900[/tex]
[tex]2 d^{2}+12 d-864=0[/tex]
Using quadratic formula, we shall determine the value of d.
[tex]d=\frac{-12 \pm \sqrt{12^{2}-4 \cdot 2(-864)}}{2 \cdot 2}[/tex]
[tex]d=\frac{-12 \pm \sqrt{144+6912}}{4}[/tex]
[tex]d=\frac{-12 \pm \sqrt{7056}}{4}[/tex]
[tex]d=\frac{-12 \pm 84}{4}[/tex]
Thus, we have,
[tex]d=\frac{-12 + 84}{4}\\d=\frac{72}{4} \\d=18[/tex] and [tex]d=\frac{-12 - 84}{4}\\d=\frac{-96}{4} \\d=-24[/tex]
Since, the value of d cannot be negative.
Thus, [tex]d=18[/tex]
Hence, the distance from the tower's base to the end of the wire is 18 feet.
Substituting [tex]d=18[/tex] in [tex](6+d)[/tex], we get,
[tex]6+18=24[/tex]
Thus, the height of the tower is 24 feet.