Question

NO LINKS!!! Please help me with this problem. PART 9

THIS IS NOT MULTIPLE CHOICE!!

a. How long should a guy wire be if it is to connect to the top of the tower and be secured at a point on the sloped side 100 feet from the base of the tower?

b. How long should a second guy wire be if it is to connect from the base on the flat side? ​

Answer 1

a) 493 ft (nearest foot)

b) 269 ft (nearest foot)

Explanation:

Part a

The guy wire creates a triangle with the tower with the following:

• longest side = 500 ft
• shortest side = 100 ft
• angle between longest side and shortest side = 90 - 10 = 80°

Therefore, as we have an angle with 2 adjacent side lengths, we can use the cosine rule to find the length of the unknown side opposite the angle.

cosine rule: c² = a² + b² - 2ab cos C

(where C is the angle, a and b are the sides adjacent the angle, and c is the side opposite the angle)

Substituting given values into the equation and solving for c:

c² = a² + b² - 2ab cos C

⇒ c² = 500² + 100² - 2(500)(100) cos(80°)

⇒ c² = 260000 - 100000cos(80°)

⇒ c = √[260000 - 100000cos(80°)]

⇒ c = 492.5801277...

Therefore, the length of the guy wire that is connected to the top of the tower and is secured at a point on the sloped side 100 ft from the base of the tower is 493 ft (nearest foot).

Part b

From inspection of the diagram, we can see that the second (left) guy wire is attached halfway up the height of the tower.

This guy wire creates a right triangle with the tower with the following:

• height = 500 ÷ 2 = 250 ft
• base = 100 ft

Therefore, as we have the length of both legs of a right triangle, we can use Pythagoras' Theorem to find the length of the wire:

Pythagoras' Theorem: a² + b² = c²

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

Substituting given values into the equation and solving for c:

a² + b² = c²

⇒ 100² + 250² = c²

⇒ c² = 72500

⇒ c = √(72500)

⇒ c = 269.2582404...

Therefore, the length of the second guy wire is 269 ft (nearest foot)

Answer 2

a. 493 ft.

b. 269 ft.

Explanation:

Please see the attached pictures.

a. Since we are interested in the length of AD, and ∆ABD does not have a right angle, we should think about using sine rule or cosine rule. We are only given two lengths in ∆ABD thus let's find one more angle, ∠ABD.

∠ABD= ∠ABC -∠CBD

∠CBD= 10° (given)

Since the tower should be vertically upright from the ground, we can assume that ∠ABC is 90°.

∠ABD= 90° -10°

∠ABD= 80°

With 2 lengths and 1 angle, we can apply cosine rule to find the length of the 3rd side:

`\textcolor{steelblue}{{ {c}^(2) = {a}^(2) + {b}^(2) - 2abcosC}}`

(AD)²= 500² +100² -2(500)(100)(cos80°)

After simplifying and taking the square root of both sides, we obtain:

AD= 492.58 (5 s.f.)

AD= 483 (3 s.f.)

Thus, the guy wire has to be
`\textcolor{red}{{483 ft}}`.

__________

b. Here, we make the assumption that the 500 ft. is excluding the height of the antenna and each square unit has the same height.

The tower is made up of 12 boxes, while the second guy wire is connected to the 6th box of the tower from the ground.

12 boxes ----- 500 ft

6 boxes ----- 500 ÷2= 250 ft

Pythagoras' Theorem states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides.

Applying Pythagoras' Theorem,

x²= 100² +250²

x²= 72500

Taking the square root of both sides:

x= 269.26 (5 s.f.)

x= 269 (3 s.f.)

Thus, the length of the second guy wire is
`\textcolor{red}{{269 ft}}`.