Easy 10 points! What is minimal distance between the cars?

Question

Two roads intersect at right angle.
First car is at intersection, driving North at speed 30mph
At this same moment the second car is 1000 feet away from the intersection, approaching from West at speed 40mph.

They did not teach us to diifferenciate yet. Our teacher says it's about frames of references.

Answer 1

Fix the first car at the intersection and consider the motion of the second car relative to the first.
It is 50 mph at an angle θ S of E
where tanθ = 3/4

From the geometry, the minimum distance is 600 feet.

Answer 2

Relative to the car which is 1000 ft away from the junction, the other car is moving .at an angle θ = 36.9° which is found from tan θ = 30/40
θ is the angle from East to North.
The shortest distance S is the perpendicular distance from the junction to the car which appears to move along this direction?
1000 f t is the hypotenuse S is the opposite side
sin 36.9°. = S ft / 1000 ft.
S = 1000 x Sin 36.9° = 600 ft
=======================

Answer 3

first write the equation
distance is square root of [ (1000-40x)^2 + (30x)^2 ]
find the extremes. start with the start and the end. x=0 and x=(1000/40).
Then you need to get the derivative and solve for all instances which it equals 0.
then just see which is lowest. not too bad really.

Answer 4

Draw a triangle with 996 feet as the base and 3 feet as the side. Then calculate the hypotenuse using:

a^2 + b^2 = c^2

Now do that for:

992/6
988/9
984/12
980/15
976/18
972/21
968/24
964/27
960/30
956/33
952/36
948/39
944/42
940/45
936/48
932/51
928/54
924/57
920/60
916/63
912/66
908/69
904/72
900/75
896/78
892/81
888/84
884/87
880/90
876/93
872/96
868/99
862/102
858/105
854/108
850/111
846/114
842/117
836/120
832/123
828/126
824/129
820/132
816/135
812/138
808/141
804/144
800/147
796/150
792/153
788/156
782/159
778/162
774/165
770/168
766/171
762/174
758/177
754/180
750/183
746/186
742/189
738/192
736/195

........Keep going until the number on the left reaches zero and the number on the right reaches 750.

Now pick out the smallest 2 values for the hypotenuse. You can get closer to the answer by subtracting 0.4 from the base and adding 0.3 to the side for all numbers in between those two smallest values and calculating the new hypotenuses.

After that, pick the smallest two values and subtract 0.04 from the base and add 0.03 to the side for all numbers in between and calculating new hypotenuses.

Then use 0.004 and 0.003, and then 0.0004 and 0.0003.

You should have an answer that's pretty close in about 5-6 hours.

Answer 5

1000 feet

Answer 6

975 feet

Answer 7

D=RT
Distance = Rate x Time