Romeo and Juliet are moving in the xy-plane. Juliet starts at the point (0,20( and moves in a straight line at a constant speed. She will pass through the origin in exactly 5 seconds. Romeo starts at the same time from the point (30,0) and moves in a straight line at a constant speed. He will pass through the origin in exactly 2 seconds. Give the parametric equations for Romeo and Juliet's location t seconds after they start moving.
Juliet
x=10-2t, y=20-4t
Romeo
x=30-15t, y=0
What is need help on is this other question. When will Romeo and Juliet be closest to each other?
Was Juliet at (0,20) (according to the question) or at (10,20) (according to the first parametric equation)?
The distance between them can be calculated by
D(t)=sqrt((x2(t)-x1(t))²+(y2(t)-y1(t))²)
By finding the derivative with respect to t and equating the resulting function to zero, t can be solved.
Hint: they would come within 13.23 units of each other (assuming Juliet was at (10,20) at the start).
To find when Romeo and Juliet will be closest to each other, we can find the point where the distance between them is minimized. The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Using the given parametric equations for Romeo and Juliet's locations, we can substitute their x and y values at any given time t:
For Juliet: xJ = 10 - 2t, yJ = 20 - 4t
For Romeo: xR = 30 - 15t, yR = 0
Now, let's find the square of the distance (to avoid taking square roots unnecessarily) between Romeo and Juliet:
Distance^2 = [(30 - 15t) - (10 - 2t)]^2 + [(0 - (20 - 4t))^2]
= [(20 - 13t)]^2 + [(4t - 20)^2]
= 169t^2 - 840t + 1000
We want to find the value of t that minimizes this equation. Since this is a quadratic equation (t^2 term present), it will have a minimum point. We can find this by finding the vertex of the quadratic.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a)
In our case, a = 169 and b = -840, so:
t = -(-840) / (2 * 169)
t = 840 / 338
t ≈ 2.4845 seconds
Therefore, Romeo and Juliet will be closest to each other approximately 2.4845 seconds after they start moving.