A standard football field is 100 yards long and 53 1/3 yards wide. A quarterback is standing on the 25 yard line, 25 yards from the left sideline, throws the ball to a receiver who is on the 35 yard line, 5 yards from the right sideline, as shown in the figure.
(a) How long was the pass? [Hint: Place the field in the first quadrant of the coordinate plane, with the left sideline on the y-axis and the goal line behind the quarterback on the x-axis. What are the coordinates of the quarterback and the receiver?] (Round your answer to one decimal place.)
(b) A player is standing halfway between the quarterback and the receiver. What are his coordinates? (Round each answer to one decimal place.:
The receiver is 10 yd downfield and 53 1/3 - 25 - 5 = 23 1/3 yds to the right.
So, the distance is clearly
√(10^2 + (70/3)^2) yds
QB: (25,25)
Rcvr: (35,48 1/3)
So, take their averages to find the midpoint.
(a) To find the length of the pass, we need to determine the distance between the quarterback and the receiver in the coordinate plane.
Given:
- Quarterback coordinates: (0, 25)
- Receiver coordinates: (35, 5)
Using the distance formula, which is the square root of the sum of the squares of the differences between x-coordinates and y-coordinates, we can calculate the length of the pass:
Length of the pass = sqrt((35-0)^2 + (5-25)^2)
Simplifying:
Length of the pass = sqrt(35^2 + (-20)^2)
Length of the pass = sqrt(1225 + 400)
Length of the pass = sqrt(1625)
Length of the pass ≈ 40.3 yards
Therefore, the length of the pass is approximately 40.3 yards.
(b) The player standing halfway between the quarterback and the receiver would have coordinates that are the average of the x-coordinates and y-coordinates of the quarterback and receiver.
Quarterback coordinates: (0, 25)
Receiver coordinates: (35, 5)
The x-coordinate of the player:
X-coordinate = (0 + 35)/2
X-coordinate = 35/2
X-coordinate = 17.5
The y-coordinate of the player:
Y-coordinate = (25 + 5)/2
Y-coordinate = 30/2
Y-coordinate = 15
Therefore, the player standing halfway between the quarterback and the receiver has coordinates (17.5, 15).
To find the length of the pass, we need to find the distance between the coordinates of the quarterback and the receiver.
(a) Let's place the football field in the first quadrant of the coordinate plane, with the left sideline on the y-axis and the goal line behind the quarterback on the x-axis.
The quarterback is standing on the 25-yard line, 25 yards from the left sideline. This can be represented by the coordinates (0,25).
The receiver is on the 35-yard line, 5 yards from the right sideline. Since the field is 53 1/3 yards wide, the right sideline would be at x-coordinate 53 1/3. Therefore, the receiver's coordinates are (53 1/3, 35).
Using the distance formula, we can calculate the length of the pass:
Length of pass = √((x2 - x1)^2 + (y2 - y1)^2)
= √((53 1/3 - 0)^2 + (35 - 25)^2)
= √((53 1/3)^2 + 10^2)
≈ √(2837.11 + 100)
≈ √2937.11
≈ 54.22 yards
Therefore, the length of the pass is approximately 54.22 yards.
(b) To find the coordinates of a player standing halfway between the quarterback and the receiver, we can take the average of their x-coordinates and y-coordinates.
x-coordinate: (0 + 53 1/3) / 2 = 53 1/6 / 2 = 26 5/6 ≈ 26.8
y-coordinate: (25 + 35) / 2 = 60 / 2 = 30
Therefore, the coordinates of the player standing halfway between the quarterback and the receiver are approximately (26.8, 30).