A radar antenna is tracking a satellite orbiting the earth. At a certain time, the radar screen shows the satellite to be 118 km away. The radar antenna is pointing upward at an angle of 58.2 ° 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.
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 horizontal and the y-axis is vertical. We can use the given information to determine the length of each component.
(a) The x-component represents the horizontal distance between the satellite and the antenna. This can be found by using the cosine function:
x = distance * cosine(angle) = 118 km * cosine(58.2°)
(b) The y-component represents the vertical distance between the satellite and the antenna. This can be found by using the sine function:
y = distance * sine(angle) = 118 km * sine(58.2°)
Now, let's calculate these values:
(a) x = 118 km * cosine(58.2°)
(b) y = 118 km * sine(58.2°)
Using a calculator, we can evaluate these expressions:
(a) x ≈ 52.43 km (rounded to two decimal places)
(b) y ≈ 98.60 km (rounded to two decimal places)
Therefore, the x-component of the position vector of the satellite relative to the antenna is approximately 52.43 km, and the y-component is approximately 98.60 km.