Question

Car A is traveling north on a straight highway and car B is traveling west on a different straight highway. Each car is approaching the intersection of these highways. At a certain moment, car A is 0.3 km from the intersection and traveling at 95 km/h while car B is 0.4 km from the intersection and traveling at 90 km/h. How fast is the distance between the cars changing at that moment? km/h

Answer 1

The rate at which the distance changes at that moment is 91.743 Km/h

Step-by-step explanation:

The time for car a to get to the intersection =
`(Distance)/(Speed)`

=
`(0.3)/(95)`

= 0.00316 h

The time for car B to get to the intersection =
`(Distance)/(Speed)`

=
`(0.4)/(90)`

= 0.00444 h

Total time =
`\sqrt{0.00316^(2) + 0.00444^(2) }`

= 0.00545 h

Total distance between the two cars at that moment =
`\sqrt{0.3^(2) + 0.4^(2) }`

= 0.5 Km

The rate at which the distance between the cars is changing =
`(0.5)/(0.00545)`

= 91.743 Km/h

Answer 2

129 km/hr

Step-by-step explanation:

Distance of Car A North of the Intersection, y=0.3km

Distance of Car B West of the Intersection, x=0.4 km

The distance z, between A and B is determined by the Pythagoras theorem

`z^2=x^2+y^2`

`z^2=0.4^2+0.3^2=0.25\\z=√(0.25)=0.5km`

Taking derivative of
`z^2=x^2+y^2`

`2z(dz)/(dt)= 2x(dx)/(dt)+2y(dy)/(dt)`

`(dx)/(dt)=90km/hr, (dy)/(dt)=95km/hr`

`2(0.5)(dz)/(dt)= 2(0.4)X90+2X0.3X95\\(dz)/(dt)=72+57=129`

The distance z, between the cars is changing at a rate of 129 km/hr.