After flying for 13 min in a wind blowing 44 km/h at an angle of 21° south of east, an airplane pilot is over a town that is 53 km due north of the starting point. What is the speed of the airplane relative to the air?
Well, I have to say, that airplane sure has some fancy moves! Flying with a wind blowing at an angle of 21° south of east? That's quite the dance! Now, let's tackle the problem.
To find the speed of the airplane relative to the air, we need to break down the velocity components. The airplane is flying due north, which means its velocity is directly northward, and the wind is blowing at an angle of 21° south of east. That means the wind's velocity has a northward component as well as an eastward component.
Now, let's do some math! The northward component of the wind's velocity is given by 44 km/h * sin(21°), and the eastward component is given by 44 km/h * cos(21°). The airplane's velocity is just the northward component since it's flying directly north.
Since the airplane flies for 13 minutes (or 0.217 hours), we can use the formula distance = velocity * time to find the distance traveled. The distance traveled is the same as the distance between the town and the starting point, which is 53 km.
So, we have the equation 53 km = velocity * 0.217 hours. We can rearrange this equation to solve for the velocity, which gives us velocity = 53 km / 0.217 hours.
So, let's plug the values in and calculate!
53 km / 0.217 hours = 244.72 km/h.
Therefore, the speed of the airplane relative to the air is approximately 244.72 km/h. That's fast enough to give even the wind a run for its money! Keep flying high, my friend!
To solve this problem, we can break it down into two components: the horizontal component and the vertical component.
Let's first focus on the horizontal component. We are given the wind speed and angle relative to east. The wind is blowing 44 km/h at an angle of 21° south of east. To find the horizontal component of the wind speed, we need to find the cosine of this angle.
cos(21°) = adjacent/hypotenuse
The adjacent side represents the horizontal component of the wind speed, and the hypotenuse represents the total wind speed. Let's calculate it:
cos(21°) = adjacent/44
Multiplying both sides by 44:
adjacent = 44 * cos(21°)
adjacent ≈ 40.80 km/h (rounded to two decimal places)
Therefore, the horizontal component of the wind speed is approximately 40.80 km/h.
Now let's calculate the actual speed of the airplane relative to the air. We know that the airplane flew for 13 minutes at this speed. Let's convert that time to hours:
13 minutes = 13/60 hours
13 minutes ≈ 0.22 hours (rounded to two decimal places)
Next, we can use the distance formula for the horizontal component:
distance = speed * time
Since the distance traveled in the horizontal direction is the same as the distance due north (53 km), we can set up the following equation:
53 km = (speed of airplane relative to air - horizontal component of wind speed) * time
Plugging in the values:
53 km = (speed of airplane relative to air - 40.80 km/h) * 0.22 hours
Let's solve for the speed of the airplane relative to the air:
53 km = (speed of airplane relative to air - 40.80 km/h) * 0.22 hours
Dividing both sides by 0.22:
speed of airplane relative to air - 40.80 km/h = 53 km / 0.22 hours
speed of airplane relative to air - 40.80 km/h ≈ 240.91 km/h
Adding 40.80 km/h to both sides:
speed of airplane relative to air ≈ 240.91 km/h + 40.80 km/h
speed of airplane relative to air ≈ 281.71 km/h (rounded to two decimal places)
Therefore, the speed of the airplane relative to the air is approximately 281.71 km/h.