1) Two boats leave the same port at the same time. One travels at a speed of 27 mi/h in the direction N 50° E and the other travels at a speed of 22 mi/h in a direction S 70° E (see the figure). How far apart are the two boats after one hour? (Round your answer to the nearest mile.)

2) A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 58°. A guy wire is to be attached to the top of the tower and to the ground, 150 m downhill from the base of the tower. The angle α in the figure is determined to be 7°. Find the length of cable required for the guy wire. (Round your answer to the nearest meter.)

Thank you!

1. x^2 = 22^2+ 27^2 -2(22)(27)cos60

x^2 =1213 -1188cos60

x^2 =619
x= 24.9 miles

#1

convert to rectangular coordinates and subtract to get a distance of 24.9

#2
assuming the angle α is the angle between the ground and the cable, then using the law of sines, the distance d of the cable is

d/sin148° = 150/sin35°

kuai's solution is better, because the assignment is evidently involving the law of sines/cosines.

also, less work!

1) Well, those boats must really be going in opposite directions to end up so far apart! It's like they're on a mission to create distance! Now, let's tackle this. We have two boats going in different directions, so we need to work out their individual distances traveled after one hour.

For the first boat, we have a speed of 27 mi/h and it's traveling in the direction N 50° E. Now, that may sound like a fancy code, but we can break it down. It means the boat is going 50° east of north. So, it's kind of like it's trying to go north but got a little sidetracked. Let's calculate how far it traveled in one hour, shall we?

Using a little trigonometry, we can find that the distance the boat traveled in one hour is 27 * sin(50°). Don't worry about the math, I've got your back! Now, let's calculate that: 27 * sin(50°) = approximately 20.77 miles.

Now, as for the other boat, it's traveling at a speed of 22 mi/h in a direction S 70° E. This boat is also veering off a little from its intended path. Let's find out how far it traveled: 22 * sin(70°) = approximately 20.66 miles.

After one hour, the two boats would be approximately 20.77 miles + 20.66 miles = approximately 41.43 miles apart. Let's round that to the nearest mile, shall we? So, the two boats would be around 41 miles apart after one hour. Wow, they really took that distance thing seriously!

2) Ah, the famous leaning tower of...communication! We have an inclined hill, an alpha angle, and a guy wire, which sounds like a really interesting mix. Let's get to work!

We have an angle of inclination of 58°, and we need to find the length of the guy wire. The guy wire will go from the top of the tower to the ground, 150 m downhill from the base of the tower, forming an angle of alpha (α) with the ground. Quite the setup!

Now, to calculate the length of the guy wire, we can use a little trigonometry magic. We have the opposite and the adjacent sides of the right triangle formed by the guy wire, so we can use the tangent function to find the length.

The length of the guy wire is given by the equation: length = opposite / tan(alpha).

Since the opposite side is the downhill distance of 150 m and alpha is 7°, we can plug in the values and crunch some numbers:
length = 150 / tan(7°) = approximately 1242.85 meters.

So, the length of the guy wire required is approximately 1242 meters. It seems like that tower is going to be pretty well-connected!

1) To find the distance between the two boats after one hour, we need to calculate their respective displacements and then find the distance between those displacements.

The first boat is traveling at a speed of 27 mi/h in the direction N 50° E. We can break down its velocity vector into its northward component and eastward component.
- The northward component = Speed * sin(angle) = 27 * sin(50°)
- The eastward component = Speed * cos(angle) = 27 * cos(50°)

So, after one hour, the first boat will have traveled a displacement of (27 * sin(50°)) miles northward and (27 * cos(50°)) miles eastward.

Similarly, for the second boat traveling at a speed of 22 mi/h in the direction S 70° E:
- The southward component = Speed * sin(angle) = 22 * sin(70°)
- The eastward component = Speed * cos(angle) = 22 * cos(70°)

After one hour, the second boat will have traveled a displacement of (22 * sin(70°)) miles southward and (22 * cos(70°)) miles eastward.

Now, we can find the difference between the northward and southward components, as well as the eastward components, to get the displacement between the two boats.

North-South displacement = (27 * sin(50°)) - (22 * sin(70°))
East-West displacement = (27 * cos(50°)) - (22 * cos(70°))

Finally, we can use the Pythagorean theorem to find the distance between the two boats:
Distance = √((North-South displacement)^2 + (East-West displacement)^2)

Calculating these values will give us the distance between the two boats after one hour.

2) To find the length of the guy wire, we can start by analyzing the right-angled triangle formed between the tower, the ground, and the guy wire.

Given that the angle of inclination of the hill is 58° and the angle α is 7°, we can deduce that the angle between the guy wire and the horizontal ground is (90° - 7°) = 83°.

Using trigonometry, we can determine the length of the guy wire as follows:

Length of guy wire / Length of hill = cos(angle between guy wire and horizontal ground)

We know the length of the hill is 150 m downhill from the base of the tower, so the length of the hill is 150 m.

Plugging in the values, we get:

Length of guy wire / 150 = cos(83°)

Rearranging the equation, we can solve for the length of the guy wire:

Length of guy wire = 150 * cos(83°)

Calculating this value will give us the length of the guy wire required.