27.1 mi / h; N 53° E
Step-by-step explanation:
Begin by finding the component form of the two vectors.
The balloon's vector has a magnitude of 20 mi/h and makes an angle of 54° with the x-axis.
The figure shows a vector of balloon speed on a Cartesian plane. The initial point of the vector is the origin. The vector of balloon speed has a length of 20 units. The angle between the vector and the positive direction of the x axis measures 54 degrees.
The balloon's vector components can be determined from the length of the hypotenuse and the angle made with the x-axis.
cos54°=x20
Multiply both sides by 20.
20(cos54°)=x
Calculate and round to the nearest tenth.
11.7≈x
sin54°=y20
Multiply both sides by 20
20(sin54°)=y
Calculate and round to the nearest tenth.
16.2≈y
The balloon's vector is <11.7,16.2>.
Since the wind blows 10 mi/h in the direction of the positive x-axis, it has a horizontal component of 10, and a vertical component of 0.
The figure shows a vector of wind speed on a Cartesian plane. The initial point of the vector is the origin. The vector of wind speed is directed along the positive x axis and has a length of 10 units.
The wind's vector is <10,0>.
To combine the vectors, add the components of the balloon's vector and the wind's vector .
<11.7,16.2>+<10,0>=<21.7,16.2>
The resultant vector in component form is <21.7,16.2>.
The magnitude of the resultant vector is the balloon's actual speed. To find the magnitude of the resultant vector, use the Distance Formula.
∣∣<21.7,16.2>∣∣=(21.7)2+(16.2)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√
Simplify.
∣∣<21.7,16.2>∣∣=470.89+262.44‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√
Simplify again.
∣∣<21.7,16.2>∣∣=733.33‾‾‾‾‾‾‾√
Simplify the square root and round to the nearest tenth.
∣∣<21.7,16.2>∣∣≈27.1 mi/h
The figure shows a vector of balloon speed, a vector of wind speed and a resultant vector on a Cartesian plane. The initial point of the vector of balloon speed is the origin. The initial point of the vector of wind speed is the terminal point of the vector of balloon speed. The initial point of the resultant vector is the origin. The terminal point of the resultant vector is the terminal point of the vector of wind speed. The angle between the resultant vector and the positive direction of x axis is labeled as A.
The angle A is formed by the resultant vector and the x-axis and gives the balloon's actual direction. The tangent of the angle formed with the x-axis is the y component over the x component.
tanA=16.221.7
Solve for m∠A
m∠A=tan−1(16.221.7)
Round to the nearest degree.
m∠A≈37°
This is equivalent to the direction N 53°E, the angle formed by the resultant vector and the y-axis.
Therefore, the balloon's actual speed is 27.1 mi/h, and actual direction is N 53°E.
