I have an assignment and I am having trouble with it. Can someone please help ASAP???
1. You are a member of your school's robotics team and are in charge of programming a robot that will pick up Ping-Pong balls. The competition arena is a rectangle with length 90 feet and width 95 feet. On graph paper, you sketch the arena as a rectangle on the coordinate plane with sides that are parallel to the coordinate axes and with the southwest corner of the arena set at the origin. Each unit width on the paper grid corresponds to 5 feet of length of the arena. You initially set the robot to move along a straight path at a constant speed. In the sketch, the robot’s position corresponds to the point (10, 30) in the coordinate plane at time = 2 seconds and to the point (40, 75) at time = 8 seconds. (Snippet of coordinate plane below)
A. Sketch the arena on the graph paper below, and write a system of inequalities that describes the region in the sketch.
B. Show that at the start, that is, at time = 0, the robot was located at a point on the west wall of the arena. How many feet from the southwest corner was it?
C. What is the speed of the robot? Round to the nearest whole number.
D. Write down an equation for the line along which the robot moves.
E. At some time the robot will hit a wall. Which wall will it hit? What are the coordinates of that point of impact?
F. How far does the robot move between time = 0 seconds and the time of this impact? What is the time for the impact? Round distance to the nearest hundredth and time to the nearest second.
At the time of impact, you have the robot come to a gentle halt and then turn and head in a direction perpendicular to the wall. Just as the robot reaches the opposite wall, it gently halts, turns, and then returns to start. (We are assuming that the robot does not slow down when it hits a wall.) The robot thus completes a journey composed of three line segments forming a triangle within the arena. Sketch the path of the robot’s motion.
G. What are the coordinates where it hits the east wall?
H. What is the perimeter of that triangle? Round to the nearest hundredth.
I. What is the area of the triangle? Round to the nearest tenth.
J. If the count of Ping-Pong balls in the arena is large and the balls are spread more or less evenly across the whole arena, what approximate percentage of balls do you expect to lie within the triangle the robot traced? (Assume the robot encountered no balls along any legs of its motion.)

A) Find the sketch in attachment.

In the sketch, we have plotted:

- The length of the arena on the x-axis (90 feet)

- The width of the arena on the y-axis (95 feet)

- The position of the robot at t = 2 sec (10,30) and its position at t = 8 sec (40,75)

The origin (0,0) is the southweast corner of the arena. The system of inequalities to descibe the region of the arena is:

B)

Since the speed of the robot is constant, it covers equal distances (both in the x- and y- axis) in the same time.

Let's look at the x-axis: the robot has covered 10 ft in 2 s and 40 ft in 8 s. There is a direct proportionality between the two variables, x and t:

So, this means that at t = 0, the value of x is zero as well.

Also, we notice that the value of y increases by (7.5 feet every second), so the initial value of y at t = 0 is:

So, the initial position of the robot was (0,15) (15 feet above the southwest corner)

C)

The speed of the robot is given by

where d is the distance covered in the time interval t.

The distance covered is the one between the two points (10,30) and (40,75), so it is

While the time elapsed is

Therefore the speed is

D)

The equation for the line of the robot is:

where m is the slope and q is the y-intercept.

The slope of the line is given by:

Which means that we can write an equation for the line as

where q is the y-intercept. Substituting the point (10,30), we find the value of q:

So, the equation of the line is

E)

By prolonging the line above (40,75), we see that the line will hit the north wall. The point at which this happens is the intersection between the lines

and the north wall, which has equation

By equating the two lines, we find:

So the coordinates of impact are (53.3, 95).

F)

The distance covered between the time of impact and the initial moment is the distance between the two points, so:

From part B), we said that the y-coordinate of the robot increases by 15 feet/second.

We also know that the y-position at t = 0 is 15 feet.

This means that the y-position at time t is given by equation:

The time of impact is the time t for which

y = 95 ft

Substituting into the equation and solving for t, we find:

G)

The path followed by the robot is sketched in the second graph.

As the robot hits the north wall (at the point (53.3,95), as calculated previously), then it continues perpendicular to the wall, this means along a direction parallel to the y-axis until it hits the south wall.

As we can see from the sketch, the x-coordinate has not changed (53,3), while the y-coordinate is now zero: so, the robot hits the south wall at the point

(53.3, 0)

H)

The perimeter of the triangle is given by the sum of the length of the three sides.

- The length of 1st side was calculated in part F:

- The length of the 2nd side is equal to the width of the arena:

- The length of the 3rd side is the distance between the points (0,15) and (53.3,0):

So the perimeter is

I)

The area of the triangle is given by:

where:

is the base (the distance between the origin (0,0) and the point (53.3,0)

is the height (the length of the 2nd side)

Therefore, the area is:

J)

The percentage of balls lying within the area of the triangle traced by the robot is proportional to the fraction of the area of the triangle with respect to the total area of the arena, so it is given by:

where:

is the area of the triangle

is the total area of the arena

Therefore substituting, we find: