Question

Rachel is trying to hang a piñata for the birthday party but doesn't know how much rope she needs. She knows the height of the tree where she needs to hang the rope is 9√5 feet tall and the angle of elevation between the ground and the rope that connects to the top of the tree is 60 degrees. What would be the length of the rope that she needs to hang the pinata? Leave in simplest radical form. feet What would be the length of the rope that she needs to hang the pinata? Round to feet the nearest tenth​

Answer 1

We can use trigonometry to solve this problem. Let the length of the rope be x. Then, we can draw a right triangle where the height of the tree is the opposite side, the length of the rope is the hypotenuse, and the angle of elevation is 60 degrees. The adjacent side can be found using the trigonometric function cosine:

cos 60° = adjacent / hypotenuse

1/2 = adjacent / x

Multiplying both sides by x, we get:

x/2 = adjacent

To find the length of the hypotenuse, we can use the Pythagorean theorem:

hypotenuse^2 = adjacent^2 + opposite^2

Substituting in the values we know, we get:

x^2 = (x/2)^2 + (9√5)^2

Simplifying the right side, we get:

x^2 = x^2/4 + 405

Multiplying both sides by 4, we get:

4x^2 = x^2 + 1620

Subtracting x^2 from both sides, we get:

3x^2 = 1620

Dividing both sides by 3, we get:

x^2 = 540

Taking the square root of both sides, we get:

x = √540 = 6√60

Therefore, the length of the rope that Rachel needs to hang the piñata is 6√60 feet. Rounded to the nearest tenth, this is approximately 74.1 feet.

Answer 2

43.4 ft

Explanation:

You want to know the length of rope needed to hang a piñata from the top of a tree that is 9√5 feet tall, if the rope makes an angle of 60° with the ground.

Triangle

The hypotenuse of the triangle can be found by considering the sine of the 60° angle. The sine of the angle is given by ...

Sin = Opposite/Hypotenuse

Then the hypotenuse of the triangle can be found from ...

sin(60°) = tree height / hypotenuse

hypotenuse = tree height/sin(60°) = 9√5/sin(60°)

Rope length

We presume the length of rope Rachel wants is the total of the lengths from the anchor point to the tree top and back to the ground. That will be ...

9√5 + 9√5/sin(60°) = 9√5·(1 +1/sin(60°)) ≈ 43.4 . . . . feet

Rachel needs about 43.4 feet of rope to hang the piñata.