Question (1 Viewer)

[letsdie45]

Hi can anyone help me with this question please?

Highway A and Highway B intersect at right angles. A car on Highway A is presently 80km from the intersection and is travelling towards the intersection at 50km per hour. A car on Highway B is presently 70km from the intersection and is travelling towards the intersection at 45km per hour.

a) Find an expression for the square of the distance between the two cars if they continue in this manner for h hours.

b) If the cars can continue through the intersection and remain on the same highways, in how many minutes will the distance between them be a minimum?

 

[Trebla]

The car on highway A travels at 50 km/h and is 80km from intersection.
The car on highway B travels at 45 km/h and is 70km from intersection.

After h hours, car on highway A travels 50h km and car on highway B travels 45h km, so distance from intersection for each are 80 - 50h and 70 - 45h respectively. Let D be distance between two cars. By Pythagoras' theorem:
D² = (80 - 50h)² + (70 - 45h)²

We want to minimise D with respect to time. This is equivalent to minimising D².
d(D²)/dh = 2(80 - 50h).(-50) + 2(70 - 45h)(-45)
= 5000h - 8000 + 4050h - 6300
= 9050h - 14300
Minimum occurs when d(D²)/dh = 0
9050h - 14300 = 0
h = 286/181
Second derivative: d²(D²)/dh² = 9050 > 0, confirms it is a local minimum. It is also the absolute minimum as there are no other turning points.
h = 286/181 hours is equivalent to 17160/181 minutes.

 

[letsdie45]

Thanks a lot, you've been very helpful.