SOLUTION: Differential calculus: Two automobiles start from point A at the same time.one travels west at 60mph and the other travels north at 35 mph. how fast is the distance between them ch

Question 246029: Differential calculus: Two automobiles start from point A at the same time.one travels west at 60mph and the other travels north at 35 mph. how fast is the distance between them changing 3 hours later.
Found 2 solutions by marcsam823, kensson:
Answer by marcsam823(57) (Show Source): You can put this solution on YOUR website!
Determine how far each car has travelled in 3 hours using
Distance = rate x time
Car A = 60mph x 3 hrs = 180 miles
Car B = 35 mph x 3 hrs = 105 miles
Draw a right triangle
Label the horizontal leg "x = 180"
Label the vertical leg "y = 105"
Label the hypotenuse "z"
Using the Pythagorean Theorem, solve for z:

Before you proceed with the calculus identify your variables:
Distances
Let x = 180
Let y = 105
Let z = 208.39
Rates of change
Let dx/dt = 60
Let dy/dt = 35
The variable you are solving for is dz/dt, the rate of change (i.e. how fast the two cars are moving apart after 3 hours)
Differentiate the Pythagorean Theorem:

Simplify the equation by dividing each term by 2

Substitute the values from above

Solve for dz/dt:

Dividing both sides by 208.39:

This is how fast the two cars are moving away from each other.

Answer by kensson(21) (Show Source): You can put this solution on YOUR website!
OK, the east-west displacement (in miles) between the cars at time t (in hours) is 60t. The north-south displacement is 35t. So the distance is by Pythagoras; that works out to be 65t.
The rate of change of the distance is the derivative of that, or 65 miles per hour.
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