Answer:
a)DL/dt is positive then L s increasing
b) L still increasing
Step-by-step explanation:
Let´s call O the place for the gas station, A the location point of police car ( at 5 miles east from the gas station ), and B the location point of the truck ( 12 miles south from a gas station.
The three points shape a right triangle with L (distance between police car and truck) then
L² = OA² + OB² at the point police car is 5 miles east and truck 12 miles south ( both from gas station)
L² = ( 5)² + (12)²
L = √ 25 + 144
L = √169
L = 13 miles
Pitagoras theorem establishes in a right triangle hypotenuse L is:
L² = a² + b² a and b are the legs then
In general
L² = x² + y ² x and y the legs over distance police-car/gas station and truck/gas station
Differentiation on both sides of the equation with respect to time give us:
2*L*DL/dt = 2*x*Dx/dt + 2*y*Dy/dt
Where Dx/dt = 130 m/h West Dy/dt = 100 m/h south
L = 13 m x = 5 m y = 12
Then by substitution
2*13*DL/dt = 2*5*130 + 2*12*100
DL/dt = (1300+ 2400) / 26
DL/dt = 142 m/h
DL/dt is positive then L s increasing
b) If truck speed is 70 m/h
DL/dt still positive but with smaller module, then L still increasing but at smaller speed
DL/dt = 1300 + 2*12*70
DL/dt = (1300 + 1680)/ 26
DL/dt = 114,6 m
