On the standard coordinate grid at initial moment, ship Tiger is at the position (0, 961), ship Lion is at the position (105, 0). Tiger sails along the straight line to the position (318, 0). At the same time, Lion sails along the straight line to the position (0, 1174). Lion will reach her destination in one hour, Tiger – in two hours.
a) Find the point of intersection of the paths of the ships.
b) At what time from the moment of departure each of the ships will pass the point of intersection of the paths.
c) At what time (in minutes after departure) the distance between Lion and Tiger will be the shortest?
d) What is the shortest distance between Lion and Tiger?
e) What will be the positions of both ships at the moment when the distance between them is the shortest?

A) (26, 882)

B) ship Tiger: Time = 0.164 hours

Ship Lion: Time = 0.257 hours

C) Ship Tiger: 0.164 × 60 = 9.84 minute

Ship Lion: 0.257× 60 =15.42minutes

D) Distance = 290.86

E) Ship Tiger (0, 961)

Ship Lion (105, 0)

Step-by-step explanation:

From the question, we can form the equation of the straight line from the initial and final position of the two ships. Using the general linear equation Y = mx + c. that is

Ship Tiger

Initial position (0, 961)

Final position (318, 0)

M = (0 - 961)/318

M = - 3.02

961 = -961/318(0) + C

C = 961. Hence the equation of the line for ship Tiger will be

Y = - 961x/318 + 961 (1)

Ship Lion

Initial position (105, 0).

Final position (0, 1174).

M = 1174/ -105 = - 11.18

0 = 1174/-105(105) + C

0 = - 1174 + C

C = 1174

The equation of the line for ship Lion will be

Y = - 1174/105(x) + 1174 (2)

At the point of intersection of the paths of the ships, they will have common Y and X. Hence equation 1 is equal to equation 2

- 961x/318 + 961 = -1174/105(x) + 1174

-961/318(x) + 1174/105(x) = 1174 - 961

8.16x = 213

X = 213/8.16 = 26.1

Substitute x in equation 1

Y = -961/318(26.1) + 961

Y = -78.88 + 961

Y = 882.12

Therefore the point of intersection of the paths of the ships is (26, 882)

B) given that Lion will reach her destination in one hour, Tiger – in two hours.

Ship Tiger

distance = root(318^2 + 961^2)

Distance = 1012.25

Speed = distance/time

Speed = 1012.25/2 = 506.12

Distance at the point of intersection will be

Distance = root(26^2 + (882-961)^2)

Distance = root(676 + 6241)

Distance = 83.12

Speed = distance/time

Time = distance /speed

Time = 83.12/506.12

Time = 0.164 hours

Ship Lion

distance = root(105^2 + 1174^2)

Distance = 1178.69

Speed = distance/time

Speed = 1178.69/1 = 1178.69

Distance at the point of intersection will be

Distance = root((26-105)^2 + (882-1174)^2)

Distance = root(6241 + 85264)

Distance = 302.5

Speed = distance/time

Time = distance /speed

Time = 302.5/1178.69

Time = 0.257 hours

C) the time in minute the distance between Lion and Tiger will be the shortest will be

Ship Tiger: 0.164 × 60 = 9.84 minute

Ship Lion: 0.257× 60 =15.42minutes

D) The shortest distance between Lion and Tiger will be achieved by using pythagorean theorem for the the distances at the point of intersection

Root (302.5^2 - 83.12^2)

Distance = 290.86

E) the positions of both ships at the moment when the distance between them is the shortest will be the initial position of both ships

That is

Ship Tiger

Initial position (0, 961)

Ship Lion

Initial position (105, 0).

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