17. Farimah and Helio are standing 15 ft. apart from each other and looking up at a kite that is with the flying between them. Farimah is flying the kite on a 57 ft. string at an angle of 68 ground. How far is Helio from the kite? 64.1 ft. 56.2 ft 60.0 ft. 53.2 ft.

Helio is 53.2 feet from the kite ⇒ the last answer

Step-by-step explanation:

* Lets change this story problem to a trigonometry problem

- Assume that there is a triangle joining between the

kite, Farimah and Helio

- The name of the triangle is KFH, where K position of the kite,

F position of Farimah and H is the position of Helio

∵ Farimah and Helio are standing 15 feet apart from each other

∴ FH = 15 feet

∵ Farimah is flying the kite on a 57 feet string at an angle

of 68 with the ground

∴ FK = 57 feet

∴ m∠KFH = 68°

∵ We need to know that Helio is how far from the kite

∴ We need to calculate the length of KH

* Now lets find the best way to find the length of KH

using the trigonometry

- We have the length of two sides and the measure of the included

angle between them, then the best way is the cosine Rule

* Lets explain the cosine rule:

- In ΔABC:

∵ a is the length of the side opposite to ∠A ⇒ a is BC

∵ b is the length of the side opposite to ∠B ⇒ b is AC

∵ c is the length of the side opposite to ∠C ⇒ c = AB

∴ a² = b² + c² - 2bc * cos (A)

∴ b² = a² + c² - 2ac * cos (B)

∴ c² = a² + b² - 2ab * cos (C)

* We will use the rule in our problem to find HK

∵ FH is k, HK is f, KF is h

∴ f² = h² + k² - 2hk * cos (F)

∵ h = 57 feet, k = 15 feet, m∠F = 68°

∴ f² = (57) ² + (15) ² - 2 (57) (15) * cos (68) = 2833.4227

∴ f = √2833.4227 = 53.2 feet

* Helio is 53.2 feet from the kite