A plane travels east from San francisco towards St. Louis at 500 miles per hour, and a turbojet travels north from new orleans toward St. louis at 600 miles per hour. Find the rate of change of the distance between them when the jets are 300 miles apart, and the plane is 100 miles from St. Louis.
Let the plane be x west, and the jet be y south of St. Louis.
Draw a diagram. Then the distance z between the two aircraft is
z^2 = x^2+y^2
At the moment in question,
300^2 = 100^2+y^2
y = 100√8
z dz/dt = x dx/dt + y dy/dt
300 dz/dt = 100(-500) + 100√8 (-600)
dz/dt ≈ -732 mi/hr
To find the rate of change of the distance between the plane and the turbojet, we can use the Pythagorean theorem. Let's assume that at a certain time t, the plane has traveled x miles from San Francisco towards St. Louis, and the turbojet has traveled y miles from New Orleans towards St. Louis.
According to the given information, the plane is traveling east at a constant speed of 500 miles per hour, and the turbojet is traveling north at a constant speed of 600 miles per hour.
Using the Pythagorean theorem, the distance between the plane and the turbojet at time t is given by:
distance^2 = (x - 100)^2 + y^2
To find the rate of change of this distance, we need to differentiate both sides of the equation with respect to time t.
Using the chain rule, the derivative of the left side is 2 * distance * (rate of change of distance), and the derivative of the right side can be calculated by differentiating each term with respect to t.
So, we have:
2 * distance * (rate of change of distance) = 2(x - 100)(rate of change of x) + 2y(rate of change of y)
Now, we are given that the distance between the plane and the turbojet is 300 miles, so we can substitute this in the equation:
2 * 300 * (rate of change of distance) = 2(x - 100)(rate of change of x) + 2y(rate of change of y)
Since the plane is traveling east and the turbojet is traveling north, the rate of change of x is 500 mph and the rate of change of y is 600 mph.
Substituting these values, we have:
600 * (rate of change of distance) = 2(x - 100)(500) + 2y(600)
Now, given that the plane is 100 miles from St. Louis, we can also substitute x = 100 into the equation:
600 * (rate of change of distance) = 2(100 - 100)(500) + 2y(600)
Simplifying further, we get:
600 * (rate of change of distance) = 2y(600)
Finally, isolating the rate of change of distance, we have:
rate of change of distance = 2y
Since y represents the distance the turbojet has traveled from New Orleans towards St. Louis, when the plane and turbojet are 300 miles apart, we can substitute y = 300 into the equation:
rate of change of distance = 2 * 300
Therefore, the rate of change of the distance between the plane and the turbojet when they are 300 miles apart is 600 mph.