A ship travels at a constant speed of 25 kilometres per hour (Kph) in a straight line from port A located at position (x_A, y_A) =(-100, -100) to port B at (x_B ,y_B )=(300 ,100).
A) Find parametric equation for the line of travel of the ship .your equation should be in terms a parameter t, and should be such that the ship is at port A when t=0 and at port B when t=1.
(b) During its journey ,the ship passes two lighthouses ,L_1 and L_2, which are located at positions (0,0) and (200.0), respectively.
write down expressions ,in terms of the parameter t of part (a) ,for the squares d_1^2 and d_2^2 of distances between the location of the ship at parameter value t and the lighthouses L_1 and L_2 respectively .Simplify your results.
please need your help.
thanks
To find the parametric equation for the line of travel of the ship, we can use the formula of a parametric equation for a straight line:
x = x_A + (x_B - x_A) * t
y = y_A + (y_B - y_A) * t
where t is the parameter that ranges from 0 to 1.
Using the given values:
x_A = -100
y_A = -100
x_B = 300
y_B = 100
Substituting these values into the equation, we get:
x = -100 + (300 - (-100)) * t
= -100 + 400 * t
= -100 + 400t
y = -100 + (100 - (-100)) * t
= -100 + 200 * t
= -100 + 200t
So, the parametric equation for the line of travel of the ship is:
x = -100 + 400t
y = -100 + 200t
To find the squares of the distances between the ship and the lighthouses L_1 and L_2, we use the distance formula:
d^2 = (x - x_lighthouse)^2 + (y - y_lighthouse)^2
For lighthouse L_1 at position (0, 0), we have:
d_1^2 = (-100 + 400t - 0)^2 + (-100 + 200t - 0)^2
= (400t - 100)^2 + (200t - 100)^2
= 160000t^2 - 80000t + 10000 + 40000t^2 - 40000t + 10000
= 200000t^2 - 120000t + 20000
For lighthouse L_2 at position (200, 0), we have:
d_2^2 = (-100 + 400t - 200)^2 + (-100 + 200t - 0)^2
= (400t - 300)^2 + (200t - 100)^2
= 160000t^2 - 240000t + 90000 + 40000t^2 - 40000t + 10000
= 200000t^2 - 280000t + 100000
Therefore, the expressions for the squares of distances are:
d_1^2 = 200000t^2 - 120000t + 20000
d_2^2 = 200000t^2 - 280000t + 100000
A) The parametric equation for the line of travel of the ship can be written as:
x(t) = x_A + (x_B - x_A) * t
y(t) = y_A + (y_B - y_A) * t
Substituting the given values, we have:
x(t) = -100 + (300 - (-100)) * t = -100 + 400t
y(t) = -100 + (100 - (-100)) * t = -100 + 200t
So, the parametric equation for the line of travel of the ship is:
x(t) = -100 + 400t
y(t) = -100 + 200t
B) To find the squares of the distances between the ship and the lighthouses L_1 and L_2, we can use the distance formula:
d^2 = (x - x_i)^2 + (y - y_i)^2
where (x, y) is the position of the ship at parameter value t, and (x_i, y_i) is the position of the lighthouse.
For L_1 at (0, 0):
d_1^2 = (x(t) - 0)^2 + (y(t) - 0)^2
= (-100 + 400t - 0)^2 + (-100 + 200t - 0)^2
= (400t - 100)^2 + (200t - 100)^2
= 160000t^2 - 80000t + 10000 + 40000t^2 - 40000t + 10000
= 200000t^2 - 120000t + 20000
For L_2 at (200, 0):
d_2^2 = (x(t) - 200)^2 + (y(t) - 0)^2
= (-100 + 400t - 200)^2 + (-100 + 200t - 0)^2
= (400t - 300)^2 + (200t - 100)^2
= 160000t^2 - 240000t + 90000 + 40000t^2 - 40000t + 10000
= 200000t^2 - 280000t + 100000
So, the expressions for the squares of the distances d_1^2 and d_2^2 are:
d_1^2 = 200000t^2 - 120000t + 20000
d_2^2 = 200000t^2 - 280000t + 100000