Question

1. use structure a 15 ft long cable is connected from a hook to the top of a pole that has an unknown height. the distance from the hook to the base of the pole is 3 ft shorter than the height of the pole. a. what can you use to find the height of the pole? b. write and solve a quadratic equation to find the height of the pole. c. how far is the hook from the base of the pole?

Explanation:

Step1: Answer part a (identify tool)

The pole, ground, and cable form a right triangle, so we use the Pythagorean theorem, which states for a right triangle with legs $a,b$ and hypotenuse $c$: $a^2 + b^2 = c^2$.

Step2: Define variables for part b

Let pole height $x$ (ft), hook distance $x-3$ (ft), cable (hypotenuse) $15$ ft.

Step3: Set up quadratic equation

Substitute into Pythagorean theorem:
$$x^2 + (x-3)^2 = 15^2$$
Expand and simplify:
$$x^2 + x^2 -6x +9 = 225$$
$$2x^2 -6x -216 = 0$$
Divide by 2:
$$x^2 -3x -108 = 0$$

Step4: Solve quadratic equation

Factor the quadratic:
$$(x-12)(x+9) = 0$$
Solutions: $x=12$ or $x=-9$. Since height cannot be negative, $x=12$.

Step5: Calculate hook distance (part c)

Substitute $x=12$ into $x-3$:
$$12 - 3 = 9$$

Answer:

a. The Pythagorean theorem
b. The quadratic equation is $x^2 - 3x - 108 = 0$, and the height of the pole is 12 ft.
c. 9 ft