Final answer:
The Cartesian point (-8,8) can be converted to polar coordinates resulting in r = -8√2 and θ = 3π/4; therefore, the correct option is b) r equals -8, theta equals 3pi/4.
Step-by-step explanation:
To convert the Cartesian point (-8,8) to polar coordinates where r is less than 0 and 0 is less than or equal to θ which is less than or equal to 2π, we must find the value of r and θ. In polar coordinates, r represents the distance from the origin to the point, and θ represents the angle the point makes with the positive x-axis.
For the point (-8,8), using Pythagorean theorem, |r| is the square root of (-8)² + (8)², which equals 8√2. Since we need r to be negative, r is -8√2. To find θ, we look at the point's location in the Cartesian plane which lies in the second quadrant where angles are between π/2 and π. The line connecting the point to the origin makes a 45-degree angle with the negative x-axis, so θ is π - π/4, which equals 3π/4. Therefore, the polar coordinates are (-8√2, 3π/4), so the correct option in the final answer is option b.