From a ship off-shore, the angle of elevation of a hill is 1.1°. After the ship moves inland at 4.5 knots for 20 min, the angle of elevation is 1.4°. How high is the hill? (1 knot = 1 nautical mile = 6080 ft per hour)
As I was just about to get the answer, I realized that it was gonna be negative, which is definitely not correct. I apologize that I can't include a picture to clarify the problem a bit. Could someone maybe please explain how to get the answer? Any help is greatly appreciated! :)
Ok, let's make a diagram showing the sideview.
Let A be the origianl position of the ship, let B be the position after 20 minutes, Let P be the top of the hill, and Q be on AB extended so that BQ and PQ form a right angle.
Summarizing:
angle Q = 90º, angle QBP = 1.4º, angle BAP = 1.1º
and AB = 4.5(20/60) = 1.5 nautical miles
By exterior angles, angle ABP = 178.6º
and angle APQ = .3 º
by sine law: BP/sin1.1 = 1.5/sin).3
5.49969 nautical miles or 33438 feet
now in right-angled triangle BQP
PQ/33438 = sin 1.4
PQ = 816.966 feet
the hill is appr. 817 feet high
No problem, I can help you solve the problem step-by-step without the need for a picture. Here's how you can find the height of the hill:
Step 1: Convert knots to nautical miles per minute
Since 1 knot = 1 nautical mile per hour, we need to convert 4.5 knots to nautical miles per minute because we're given a time in minutes. To do this, divide 4.5 by 60 (because there are 60 minutes in an hour):
4.5 knots ÷ 60 min = 0.075 nautical miles per minute
Step 2: Calculate the horizontal distance the ship moved inland
To find the horizontal distance the ship moved inland, we need to multiply the speed in nautical miles per minute by the time in minutes. From Step 1, we have the speed in nautical miles per minute as 0.075, and the time given is 20 minutes, so:
0.075 nautical miles per minute × 20 min = 1.5 nautical miles
Step 3: Use trigonometry to find the height of the hill
Let's assume the height of the hill is "h". We can set up the following equation using trigonometry:
tan(1.4°) = h / 1.5
Step 4: Solve for the height of the hill
Rearrange the equation to solve for "h":
h = 1.5 × tan(1.4°)
Step 5: Calculate the height of the hill
Using a calculator, evaluate the expression:
h ≈ 0.036 nautical miles
Step 6: Convert nautical miles to feet
Since we want the height of the hill in feet, multiply the result from Step 5 by the conversion factor of 6080 feet per nautical mile:
0.036 nautical miles × 6080 ft/nautical mile = 219.648 ft
The height of the hill is approximately 219.648 feet.
No problem, I can help explain how to solve this problem.
To find the height of the hill, we can use the concept of trigonometry and the relationships between angles, distances, and height. Here's the step-by-step process:
1. Convert the speed of the ship from knots to feet per minute.
- 1 knot = 6080 ft per hour, which equals 6080/60 = 101.33 ft per minute.
- So, the speed of the ship is 4.5 knots x 101.33 ft per minute = 456 ft per minute.
2. Convert the time from minutes to hours.
- 20 minutes ÷ 60 = 0.33 hours.
3. Calculate the distance the ship moves inland.
- Distance = Speed x Time
- Distance = 456 ft per minute x 0.33 hours = 150.48 ft.
4. Calculate the tangent of the initial angle of elevation.
- Tangent(1.1°) = height of the hill / distance from the ship to the hill.
- Let's denote the height of the hill as h1.
- So, tangent(1.1°) = h1 / 0 ft (since the ship is still offshore).
5. Calculate the tangent of the angle of elevation after the ship moves inland.
- Tangent(1.4°) = height of the hill / (distance from the ship to the hill + 150.48 ft).
- Let's denote the height of the hill after the ship moves inland as h2.
- So, tangent(1.4°) = h2 / 150.48 ft.
6. Solve the two equations simultaneously to find the height of the hill.
- Since tangent(1.1°) = h1 / 0 = 0, we can ignore this equation.
- Thus, tangent(1.4°) = h2 / 150.48 ft.
- Rearranging the equation, we have h2 = tangent(1.4°) x 150.48 ft.
7. Calculate the height of the hill.
- Using a calculator, find the tangent of 1.4° and multiply it by 150.48 ft.
- This will give you the height of the hill in feet.
And that's how you can solve this problem and find the height of the hill. Remember to round your final answer to an appropriate number of decimal places based on the given measurements and calculations.