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Question 1081529: Kirby moves with constant speed 5 units per hour along the line  crossing
the y-axis at midnight and the x-axis later. When is the x-axis crossing made? What does it
mean to say that Kirby’s position is a function of time? What is Kirby’s position 1.5 hours
after midnight? What is Kirby’s position t hours after midnight?
Found 2 solutions by Theo, Alan3354:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! kirby moves at a constant speed of 5 units per hour along the line y = 3x/4 + 6
the equation of the line can be written as y = 3/4 * x + 6
this is the general form of y = mx + b
the slope is 3/4 and the y-intercept is 6.
the convention is that the slope measures the movement along the line going from left to right.
if you are moving along the line from right to left, then the movement is reversed.
for example:
assuming your slope is 3/4 and you are moving along the line from the left side of the graph to the right side of the graph, which is the norm, then
you move up 3 units along the y-axis for every 4 units you move to the right along the x-axis.
however, assuming your slope is 3/4 and you are moving along the line from the right side of the graph to the left side of the graph, then you move down 3 units along the y-axis for every 4 units you move to the left along the x-axis.
your equation is y = 3/4 * x + 6
to find your y-intercept, make the value of x = 0 and solve for y.
you will get y = 3/4 * 0 + 6 which results in y = 6.
your y-intercept is at y = 6 which is the coordinate point of (0,6).
to find your x-intercept, make the value of y = 0 and solve for x.
you will get 0 = 3/4 * x + 6.
when you solve for x, you will find that your x-intercept is at x = -8 which is the coordinate point of (-8,0).
the following graph of your equation shows that this is true.
for every hour, you move 5 units along the line.
since your slope is 3/4, this means that, for every hour, you move 3/5 * 5 units = 3 units along the y-axis and 4/5 * 5 = 4 units along the x-axis.
if you are moving from left to right on the graph, then you are moving up 3 units up and 4 units to the right from your previous position.
if you are moving from right to left on the graph, then you are moving down 3 units and 4 units to the left from your previous position.
based on the picture of the graph i showed you above, you will be moving from the right side of the graph to the left side of the graph along the line.
that means, for every hour, you move 5 units along the line from the rightmost point along the line to the leftmost point along the line.
that also means, for every hour, you move 3 units down along the y-axis and 4 units to the left along the x-axis.
if you look at the y-intercept point and the x-intercept point, you will find that you are going from (0,6) to (-8,0)
the distance you travel along the line is the square root of (the square of the change in your x-coordinate value plus the square of the change in your y-coordinate value).
that becomes the square root of (64 + 36) = square root of 100 = 10.
the distance along the line from your y-intercept to your x-intercept is 10 units.
since you move 5 units per hour along the line, that means it takes 2 hours to get from the y-intercept to the x-intercept.
in 2 hours, your downward movement along the y-axis is 3 * 2 = 6
in 2 hours, your movement to the left along the x-axis is 4 * 2 = 8.
since you started from (0,6), you would wind up at (-8,0).
your movement along the line is 5 units per hour going from the right side of the graph to the left.
your movement along the y-axis is be 3 units down every hour.
your movement along along the x-axis is be 4 units to the left every hour.
if you allow the number of hours you travel to be equal to t, then your formula would be:
distance along the line is 5 * t moving from the right side of the graph to the left side of the graph.
distance along the y-axis would be 3 * t moving down along the y-axis.
distance along the x-axis would be 4 * t moving to the left along the x-axis.
you would use this formula as follows:
assume 2 hours travel time.
distance along the line is 10 units when you travel from the right side of the graph to the left side of the graph.
distance down along the y-axis is 3 * 2 = 6 units down.
distance to the left along the x-axis is 4 * 2 = 8 units to the left.
the new point is (0,6) minus (8,6) = (-8,0) which we already know is correct, based on what we see on the first graph shown.
if she travels for 1.5 hours from midnight, then:
the distance along the line would be 7.5 units traveling from the right side of the graph to the left side.
distance down along the y-axis would be 3 * 1.5 = 4.5 units.
distance to the left along the x-axis would be 4 * 1.5 = 6 units.
the new position would be (0,6) minus (6,4.5) = (-6,1.5)
this new position can be seen on the following graph.
your distance traveled in a downward direction is (6-1.5) = 4.5 units
your distance traveled in a right to left direction is (0-(-6)) = 6 units.
your distance traveled along the line is the hypotenuse of the right triangle formed which is equal to square root of (4.5^2 + 6^2) which is equal to square root of (20.25 + 36) which is equal to square root of 56.25 which is equal to 7.5.
everything checks out so we can derive the final formula you need as follows:
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the x-coordinate of kirby's position t hours after midnight would be 0 minus 4 * t
the y-coordinate of kirby's position t hours after midnight would be 6 minus 3 * t
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when t = 1.5 after midnight, you get:
x-coordinate of the new position is 0 - (4 * 1.5) = 0 - 6 = -6.
y-coordinate of the new position is 6 - (3 * 1.5) = 6 - 4/5 = 1.5.
the coordinate of the new position is (-6,1.5)
this agrees with what is shown on the second graph above.
the general formula works for any number of hours.
movement along the line is in a right side of the graph to the left side of the graph direction.
since the line is sloping up in a left side of the graph to the right side of the graph direction, then the line is sloping down in a right side of the graph to the left side of the graph direction.
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! Kirby moves with constant speed 5 units per hour along the line crossing
the y-axis at midnight and the x-axis later. When is the x-axis crossing made?
The x-axis is crossed when y=0.
y = 3x/4 +6 = 0
3x = -24
x = -8
The distance from (-8,0) to (0,6) = 10 units.
10/5 = 2 hours --> x-axis crossed at 0200.
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What does it mean to say that Kirby’s position is a function of time?
It varies with time.
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What is Kirby’s position 1.5 hours after midnight?
He's 7.5 units from (0,6) on the line, and 2.5 units from (-8,0)
--> (-6,1.5)
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What is Kirby’s position t hours after midnight?
Since he's moving right to left (on the graph):
x(t) = -t
y(t) = -3t/4 + 6
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