Consider the Sun to be at the origin of an xy-coordinate system. A telescope spots an asteroid in the xy-plane at a position given by (x, y) with a velocity given by (vx, vy).

[DATA: (x, y) = (3.3, 5.2) ×10^11 m;
(vx, vy) = (-6.1, -4.0) ×10^3 m/s.]

(A) What will the asteroid's speed be at closest approach (perihelion)?

(B) What will the asteroid's distance from the Sun be at closest approach?

To answer these questions, we need to use the position and velocity information given for the asteroid. The speed of the asteroid can be calculated using the components of its velocity, and the distance from the Sun can be determined using the coordinates of its position.

(A) To find the asteroid's speed at closest approach (perihelion), we need to calculate its magnitude of velocity. The magnitude of the velocity can be determined using the Pythagorean theorem.

Magnitude of velocity = √(vx^2 + vy^2)

Given the values:
vx = -6.1 × 10^3 m/s
vy = -4.0 × 10^3 m/s

Substituting these values into the equation:

Magnitude of velocity = √((-6.1 × 10^3)^2 + (-4.0 × 10^3)^2)

Calculating this expression will give you the magnitude of the velocity, which represents the speed of the asteroid at closest approach.

(B) To find the distance from the Sun at closest approach, we can use the distance formula to calculate the magnitude of the position vector of the asteroid.

Distance from Sun = √(x^2 + y^2)

Given the values:
x = 3.3 × 10^11 m
y = 5.2 × 10^11 m

Substituting these values into the equation:

Distance from Sun = √((3.3 × 10^11)^2 + (5.2 × 10^11)^2)

Calculating this expression will give you the magnitude of the distance from the Sun to the asteroid at closest approach.

Please note that the final answers will depend on the specific values of the coordinates and velocities provided in the data.