QUESTION 1
Joe intercepts a pass 8 yards into the end zone, 19 yards from the right side of the field as he faces the opposite end of the field (see figure). He starts running diagonally, moving at a rate of 4 yd/s down the field and 2 yd/s across the field. Let x and y be his coordinates on the field.
Write parametric equations for his x- and y-coordinates as functions of time t, in seconds, since he intercepted the ball.
If he keeps going straight at the same velocity, what will be his position when t = 10 s?
Yard Line:
Distance from the right side line:
If he keeps going at the same velocity, at what time will he cross the opponent's 40-yard line, x = 60?
The football field is 53 1/3 yards wide. when he crosses the opponent's 40-yard line, will he still be in bounds?
Diagram can be found blondebeliever.tumblr.[com]/precalc (on my blog) under question 1!
I don't have the diagram. If we call (0,0) the goal line at the edge of the field he's 19 yds from, and +x is downfield, and +y is across the field,
x(t) = -8 + 4t
y(t) = 19 + 2t
x(10) = 32
y(10) = 39
60 = -8 + 4t
t = 17
y(17) = 19 + 34 = 53. He'll be a foot inside the sideline. 1ST DOWN!
To find the parametric equations for Joe's x- and y-coordinates as functions of time t, we can use his velocity components in the x- and y-directions.
Given:
Initial x-coordinate: 19 yards
Initial y-coordinate: -8 yards
Rate of change in the x-coordinate: 2 yd/s
Rate of change in the y-coordinate: 4 yd/s
The parametric equations for his x- and y-coordinates can be written as follows:
x = 19 + 2t
y = -8 + 4t
Now, let's find Joe's position when t = 10 seconds:
To find his x-coordinate at t = 10 seconds, substitute t = 10 into the x-coordinate equation:
x = 19 + 2(10) = 19 + 20 = 39 yards
To find his y-coordinate at t = 10 seconds, substitute t = 10 into the y-coordinate equation:
y = -8 + 4(10) = -8 + 40 = 32 yards
Therefore, when t = 10 seconds, Joe's position will be at (39, 32) on the field.
To determine the yard line and distance from the right side line when Joe crosses the opponent's 40-yard line (x = 60), we can set the x-coordinate equation equal to 60 and solve for t:
60 = 19 + 2t
2t = 60 - 19
2t = 41
t = 41/2
t = 20.5 seconds
Therefore, Joe will cross the opponent's 40-yard line at t = 20.5 seconds.
To check if he will still be in bounds when he crosses the opponent's 40-yard line, we need to determine if his y-coordinate is within the width of the football field, which is 53 1/3 yards.
Substituting t = 20.5 into the y-coordinate equation:
y = -8 + 4(20.5) = -8 + 82 = 74 yards
Since Joe's y-coordinate of 74 yards is within the width of the football field (53 1/3 yards), he will still be in bounds when he crosses the opponent's 40-yard line.