Let's break down each part of the problem and go through the steps to solve it correctly. Starting with part A:
To find the plane's altitude after 1 minute, you need to use trigonometry and the given angle of ascent. Here's how to do it:
Step 1: Convert the given angle from degrees to radians.
To convert degrees to radians, remember that π radians is equal to 180 degrees. Divide the given angle (18 degrees) by 180 and then multiply by π to obtain the equivalent angle in radians.
18 degrees = (18/180) * π = 0.1π radians
Step 2: Use trigonometry to find the vertical component of the plane's speed.
The vertical component can be found by multiplying the total speed (275 feet per second) by the cosine of the given angle (18 degrees), which you've correctly done. So the vertical component is:
275 * cos(18 degrees) = 275 * cos(0.1π radians) ≈ 261.5 feet per second
Step 3: Multiply the vertical component of the speed by the time elapsed (1 minute = 60 seconds) to find the altitude.
Altitude = Vertical component of speed * Time elapsed
Altitude = 261.5 feet per second * 60 seconds = 15,690 feet
Therefore, the correct answer for part A is that the plane's altitude after 1 minute is 15,690 feet.
Now let's move on to part B:
To determine how long it takes for the plane to climb to an altitude of 10,000 feet, we need to use similar concepts and calculations.
Step 1: Set up the trigonometric equation based on the given information.
We know that the vertical displacement (change in altitude) is 10,000 feet. We need to find the time it takes for the plane to reach this altitude. Using trigonometry, we can set up the equation:
Vertical displacement = Vertical component of speed * Time elapsed
10,000 feet = 275 * cos(18 degrees) * Time elapsed
Step 2: Solve for Time elapsed.
Divide both sides of the equation by 275 * cos(18 degrees) to isolate the Time elapsed:
Time elapsed = 10,000 feet / (275 * cos(18 degrees))
Calculating this expression, we find:
Time elapsed ≈ 37.52 seconds
Therefore, the correct answer for part B is that it will take approximately 37.52 seconds for the plane to climb to an altitude of 10,000 feet.
It seems there might have been some errors in your previous calculations. By following these steps, you should arrive at the correct answers.