A radar antenna is tracking a satellite orbiting the earth. At a certain time, the radar screen shows the satellite to be 181 km away. The radar antenna is pointing upward at an angle of 66.2 ° above the ground. Find (a) the x and (b) the y components (in km) of the position vector of the satellite, relative to the antenna.
a. X = 181*Cos66.2
b. Y = 181*sin66.2
Well, let's get to the math! To find the x and y components of the position vector, we'll need to use some trigonometry. Are you ready to have some fun with numbers? Let's go!
First, let's start by defining the x and y axes. The x-axis will point horizontally, and the y-axis will point vertically. Since the radar antenna is pointing upward at an angle of 66.2°, the y component of the position vector will be positive, while the x component will be zero.
To find the y component, we can use the sine of the angle. The formula to find the y component is: y = distance * sin(angle).
In this case, the distance is 181 km, and the angle is 66.2°. So, let's calculate!
y = 181 km * sin(66.2°)
Now, I'm no mathematician, but I can certainly tell you that the answer to this equation is somewhere between "Eureka!" and "I ran out of fingers to count on." Take your time and give it a go!
And remember, if you ever get stuck, I'm here to crack a joke and lighten the mood. Keep up the good work!
To find the x and y components of the position vector of the satellite relative to the antenna, we can use trigonometry.
Let's assume the x-axis represents the horizontal direction and the y-axis represents the vertical direction, with the origin at the radar antenna.
Step 1: Find the x-component:
The x-component represents the horizontal displacement of the satellite relative to the antenna.
We can find it by using the formula: x = distance * cos(angle)
Given:
Distance = 181 km
Angle = 66.2°
Calculating the x-component:
x = 181 km * cos(66.2°)
= 181 km * 0.406
= 73.686 km (rounded to three decimal places)
So, the x-component of the position vector of the satellite relative to the antenna is approximately 73.686 km.
Step 2: Find the y-component:
The y-component represents the vertical displacement of the satellite relative to the antenna.
We can find it by using the formula: y = distance * sin(angle)
Given:
Distance = 181 km
Angle = 66.2°
Calculating the y-component:
y = 181 km * sin(66.2°)
= 181 km * 0.914
= 165.734 km (rounded to three decimal places)
So, the y-component of the position vector of the satellite relative to the antenna is approximately 165.734 km.
Therefore, the x-component is approximately 73.686 km, and the y-component is approximately 165.734 km.
To find the x and y components of the position vector of the satellite relative to the antenna, we can use trigonometry. Let's assume that the x-axis is parallel to the ground and the y-axis is perpendicular to the ground, pointing upward.
(a) To find the x-component:
We can use the cosine function to determine the x-component of the position vector. The angle between the position vector and the x-axis is the same as the angle of elevation, 66.2°.
Using cosine:
cos(angle) = adjacent/hypotenuse
cos(66.2°) = x-component/181 km
Rearranging the equation:
x-component = 181 km * cos(66.2°)
x-component ≈ 65.848 km
So, the x-component of the position vector of the satellite relative to the antenna is approximately 65.848 km.
(b) To find the y-component:
We can use the sine function to determine the y-component of the position vector. Again, the angle between the position vector and the y-axis is the same as the angle of elevation, 66.2°.
Using sine:
sin(angle) = opposite/hypotenuse
sin(66.2°) = y-component/181 km
Rearranging the equation:
y-component = 181 km * sin(66.2°)
y-component ≈ 169.77 km
So, the y-component of the position vector of the satellite relative to the antenna is approximately 169.77 km.
Therefore,
(a) The x-component is approximately 65.848 km.
(b) The y-component is approximately 169.77 km.