Question

A chameleon is resting quietly on a lanai screen, waiting for an insect to come by. Assume the origin of a Cartesian coordinate system at the lower left-hand corner of the screen and the horizontal direction to the right as the +x-direction. If its coordinates are (2.000 m, 1.000 m), (a) how far is it from the corner of the screen? (b) What is its location in polar coordinates?

a) (a) 1.414 m, (b) (1.414 m, 45°)
b) (a) 3.000 m, (b) (3.000 m, 30°)
c) (a) 2.236 m, (b) (2.236 m, 60°)
d) (a) 2.500 m, (b) (2.500 m, 45°)

Answer 1

The chameleon is 2.236 meters away from the corner of the screen, and its polar coordinates are approximately (2.236 m, 63°), calculated using the Pythagorean theorem and arctangent function.

Step-by-step explanation:

The student wants to know how far the chameleon is from the corner of a screen and its polar coordinates given its Cartesian coordinates (2.000 m, 1.000 m). To find the distance from the origin, we use the Pythagorean theorem, which gives us the radius in polar coordinates:

r = √(x² + y²) = √(2.000² + 1.000²) = √(4.000 + 1.000) = √5 ≈ 2.236 m

To find the angle θ in polar coordinates, we use the arctan function:

θ = arctan(y/x) = arctan(1.000/2.000) = arctan(0.5) ≈ 26.57° = 26.57°

However, this angle is between the line and the positive x-axis. To find the angle measured counter-clockwise from the positive x-axis (standard polar coordinates convention), we use 90° - 26.57° = 63.43°, which rounds to approximately 63°. This is due to the screen being positioned with the x-axis horizontally and to the right.

Therefore, the correct answer is (c), which shows the chameleon is (a) 2.236 m away from the corner of the screen, and (b) at the location (2.236 m, 63°) in polar coordinates.

Answer 2

b

Step-by-step explanation: