The distance d is 9 ft and the height is 12ft.
Here we can model the situation with a right triangle, where the length of the wire is the hypotenuse.
The height is one cathetus and the distance is the other catheti.
Let's define:
We know that the height of the tower is 3 ft larger than the distance, then:
h = d + 3ft
Now we can use the Pythagorean theorem, it says that the sum of the squares of the cathetus is equal to the square of the hypotenuse.
Then:
[tex]d^2 + (d + 3ft)^2 = (15ft)^2[/tex]
Now we can solve this equation for d:
[tex]d^2 + d^2 + 6ft*d + 9ft^2 = (15ft)^2\\\\2d^2 + 6ft*d - 216 ft^2 = 0\\\\d^2 + 3ft*d - 108ft^2 = 0[/tex]
Then the solutions are:
[tex]d = \frac{-3ft \pm \sqrt{(3ft)^2 - 4*(-108ft^2)} }{2} \\\\d = \frac{-3ft \pm 21ft }{2}[/tex]
We only take the positive solution:
d = (-3ft + 21ft)/2 = 9ft
And the height is 3 ft more than that, so:
h = 9ft + 3ft = 12ft
The distance d is 9 ft and the height is 12ft.
If you want to learn more about right triangles:
https://brainly.com/question/2217700
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