Question

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?

Answer 1

We need x

Apply Pythagorean theorem

`\\ \rm\Rrightarrow 15^2-x^2=(x-3)^2`

`\\ \rm\Rrightarrow 225-x^2=x^2-6x+9`

`\\ \rm\Rrightarrow 2x^2-6x-216=0`

`\\ \rm\Rrightarrow x^2-3x-108=0`

`\\ \rm\Rrightarrow (x+9)(x-12)=0`

`\\ \rm\Rrightarrow x=-9,12`

• Distance must be positive so x=12

Answer 2

a) Pythagoras' Theorem: a² + b² = c²

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

b) Given:

• a = (x - 3)
• b = x
• c = 15

Substituting these values into the equation and rearranging:

⇒ (x - 3)² + x² = 15²

⇒ x² - 6x + 9 + x² = 225

⇒ 2x² - 6x - 216 = 0

c) Solve 2x² - 6x - 216 = 0 by factorizing:

2x² - 6x - 216 = 0

⇒ x² - 3x - 108 = 0

⇒ x² - 12x + 9x - 108 = 0

⇒ x(x - 12) + 9(x - 12) = 0

⇒ (x - 12)(x + 9) = 0

⇒ x - 12 = 0 ⇒ x = 12

⇒ x + 9 = 0 ⇒ x = -9

As distance is positive, x = 12 only.

Therefore, the height of the pole is 12 ft.

So the distance from the hook to the base = 12 - 3 = 9 ft