Math  /  Trigonometry

Question4. An airplane is travelling at 500 km/h500 \mathrm{~km} / \mathrm{h} due south when it encounters a win from W45NW 45^{\circ} \mathrm{N} at 100 km/h100 \mathrm{~km} / \mathrm{h}. a. What is the resultant velocity of the airplane? b. How long will it take for the airplane to travel 1000 km ?

Studdy Solution

STEP 1

1. The airplane's velocity is 500km/h 500 \, \text{km/h} due south.
2. The wind's velocity is 100km/h 100 \, \text{km/h} from W45N W 45^\circ N , which means it is blowing towards E45S E 45^\circ S .
3. The airplane's resultant velocity is the vector sum of its velocity and the wind's velocity.

STEP 2

1. Decompose the wind's velocity into its southward and eastward components.
2. Determine the resultant velocity of the airplane by vector addition.
3. Calculate the magnitude of the resultant velocity.
4. Determine the time taken to travel 1000km 1000 \, \text{km} using the resultant velocity.

STEP 3

Decompose the wind's velocity into components. Since the wind is blowing from W45N W 45^\circ N , its components are:
- Southward component: 100cos(45)km/h 100 \cos(45^\circ) \, \text{km/h} - Eastward component: 100sin(45)km/h 100 \sin(45^\circ) \, \text{km/h}
Southward component=100×22=502km/h \text{Southward component} = 100 \times \frac{\sqrt{2}}{2} = 50\sqrt{2} \, \text{km/h} Eastward component=100×22=502km/h \text{Eastward component} = 100 \times \frac{\sqrt{2}}{2} = 50\sqrt{2} \, \text{km/h}

STEP 4

Determine the resultant velocity of the airplane by adding the components:
- Southward resultant: 500+502km/h 500 + 50\sqrt{2} \, \text{km/h} - Eastward resultant: 502km/h 50\sqrt{2} \, \text{km/h}

STEP 5

Calculate the magnitude of the resultant velocity using the Pythagorean theorem:
Vresultant=(500+502)2+(502)2 V_{\text{resultant}} = \sqrt{(500 + 50\sqrt{2})^2 + (50\sqrt{2})^2}
Calculate:
Vresultant=(500+70.71)2+70.712 V_{\text{resultant}} = \sqrt{(500 + 70.71)^2 + 70.71^2} 570.712+70.712 \approx \sqrt{570.71^2 + 70.71^2} 325709.5041+4999.9041 \approx \sqrt{325709.5041 + 4999.9041} 330709.4082 \approx \sqrt{330709.4082} 575.12km/h \approx 575.12 \, \text{km/h}

STEP 6

Determine the time taken to travel 1000km 1000 \, \text{km} using the resultant velocity:
Time=DistanceVelocity=1000km575.12km/h \text{Time} = \frac{\text{Distance}}{\text{Velocity}} = \frac{1000 \, \text{km}}{575.12 \, \text{km/h}}
Calculate:
Time1000575.12 \text{Time} \approx \frac{1000}{575.12} 1.74hours \approx 1.74 \, \text{hours}
The resultant velocity of the airplane is approximately 575.12km/h \boxed{575.12 \, \text{km/h}} and it will take approximately 1.74hours \boxed{1.74 \, \text{hours}} to travel 1000km 1000 \, \text{km} .

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