Question

Car b is 250 miles east from car a, car b is traveling west 60mph while car a is traveling north 80mph. What is the minimum distance between the two cars? What time does the minimum distance occur?​

Answer 1

Answer:

d = 200 miles

t = 1.5 hrs after start

Step-by-step explanation:

Set the origin at car a starting point.

equation for distance to car b

x = 250 - 60t

equation for distance to car a

y = 80t

distance between car a and car b

d = √((250 - 60t)² + (80t)²)

d = √(62500 - 30000t + 3600t² + 6400t²)

d = √(62500 - 30000t + 10000t²)

d = 100√(6.25 - 3t + t²)

setting the derivative to zero

0 = 100t - 150 / √(t² - 3t - 6.25)

The denominator cannot be zero, so the numerator must

0 = 100t - 150

150 = 100t

t = 1.5 hrs

d = √((250 - 60(1.5))² + (80(1.5))²)

d = √(160² + 120²)

d = 200 miles