Trigonometry Help?

Question

I need help with this word problem, that I cannot figure out how to solve. Here it is:
Each side of a tent forms a right angle with the ground. The tops of 2 ropes are attached to each side of the tent 8 ft above the ground. The other ends of the 2 ropes are attached to stakes on the ground. If the rope is 12 ft long, what angle does it make with the level ground? And what is the distance between the botton of the tent and each stake?

Answer 1

Draw a right-angled triangle: the base is the ground, with a vertical line forming the side of the tent, and the rope is the hypotenuse.

The ropes are attached 8 ft above the ground, so the height of the vertical side is 8 ft.
The rope is 12 ft long, so the hypotenuse is 12 ft.

The distance between the bottom of the tent and the stake is the third side of the triangle, so by Pythagoras this is √(12^2 - 8^2) = √(80) = 4√5 ≈ 8.94 ft.

The angle with the ground is the angle opposite the vertical side, say θ. It has opposite side length 8 ft and hypotenuse 12 ft => sin θ = 8/12 = 2/3. So the angle is arcsin (2/3) ≈ 41.8°.

Answer 2

I hope you have sketched it on paper.
Lets do it like this

Let the distance from the stake to the tent be side A,
the side of the tent be side B,
and the rope be side C,

According to pythagorus theorem

A^2 + B^2 = C^2

We are looking for the distance of A
So we make A the subject of the formula

A^2 + B^2 = C^2
A^2 = C^2 - B^2
A = square root of (C^2 - B^2)

C = 12 ft
B = 8 ft

A = squareroot (12^2 - 8^2)
= squareroot (144 - 64)
= squareroot (80) ft

The squareroot of 80 in ft is the answer. I dont have a calculator so work it out. It should be 8 point something ft though, almost 9ft.
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TO FIND THE ANGLE

Lets name angle the rope makes with the ground X

Sin = opposite/ hypotenus
= B/C

sin X = 8/12
= 2/3

X = sin ^ -1 (2/3)

OR

X = Sin ^ -1 ( 0. 6667)

Thats the answer. Again, use you calculator cos I dont have one rigt now

Answer 3

Forget the fact that there are two poles and two ropes. Just look at one of them.

You have a tent pole which is 8 feet high. You have flat ground. You have the rope coming down from the top of the pole in a straight line to the ground. The rope, the pole and the ground form a right angled triangle, because the ground is level and the pole is vertical.

Which side of the triangle is the hypotenuse?

What is the length of the hypotenuse.

Which sides of the triangle are the legs?

What do you know about the lengths of the sides of a right angled triangle?

Which sides to you know the length of?

Now you should be able to work it out.

Answer 4

Draw a right angled triangle with vertical arm(P)= 8 & hypotenuse(H)=12.
base^2 = H^2-P^2= 144-64=80
base= 80^.5= 8.94427 ft.= dist of tent base from stake.

sin x = p/h= 8/12=2/3
sin(inv)x = 41.81 deg= angle of rope with ground