Answer:
To solve this problem, we can use trigonometric concepts to find the eastward and northward components of the plane's velocity.
Given:
Speed of the plane = 225 mph
Direction of the plane = 50°
Step-by-step explanation:
To solve this problem, we can use trigonometric concepts to find the eastward and northward components of the plane's velocity.
Given:
Speed of the plane = 225 mph
Direction of the plane = 50°
(a) To find how far east of its starting point the plane is after half an hour:
Since the plane travels at a constant speed of 225 mph, in 0.5 hours (which is half an hour), it covers a distance of
225
×
0.5
=
112.5
225×0.5=112.5 miles.
Now, we need to find the eastward component of this distance. To do this, we use trigonometric functions. The eastward component is the adjacent side of the angle.
Eastward distance
=
Distance
×
cos
(
angle
)
Eastward distance=Distance×cos(angle)
Eastward distance
=
112.5
×
cos
(
50
°
)
≈
112.5
×
0.6428
≈
72.071
miles
Eastward distance=112.5×cos(50°)≈112.5×0.6428≈72.071 miles
(b) To find how far north of its starting point the plane is after 2 hours and 17 minutes:
First, we need to convert 2 hours and 17 minutes to hours. There are 60 minutes in an hour, so 17 minutes is
17
/
60
=
0.2833
17/60=0.2833 hours.
The total time is 2 hours + 0.2833 hours = 2.2833 hours.
Now, we can find the total distance traveled in 2.2833 hours:
Distance
=
Speed
×
Time
=
225
×
2.2833
≈
514.9925
miles
Distance=Speed×Time=225×2.2833≈514.9925 miles
Now, we need to find the northward component of this distance. The northward component is the opposite side of the angle.
Northward distance
=
Distance
×
sin
(
angle
)
Northward distance=Distance×sin(angle)
Northward distance
=
514.9925
514.9925×0.766
≈394.322
miles
Northward distance=514.9925×sin(50°)≈514.9925×0.766≈394.322 miles
So, after half an hour, the plane is approximately 72.1 miles east of its starting point. After 2 hours and 17 minutes, it's approximately 394.3 miles north of its starting point.
