Two cars start moving from the same point. One travels south at 60 mph and the other travels west at 25 mph. At what rate is the distance between the cars increasing 2 hours later?
Let x = the distance covered by the south traveling car.
Let y = the distance covered by the west traveling car.
Let z = the distance between the cars.
In this problem you are given two rates. What are they? Express your answers in the form dx/dt, dy/dt, or dz/dt = a number. Enter your answers in the order of the variables shown; that is, dx/dt first, dy/dt, etc. next.
What rate are you trying to find?
Write an equation relating x and y . Note: In order for WeBWorK to check your answer you will need to write your equation so that it has no denominators. For example, an equation of the form 2/x = 6/y should be entered as 6x=2y or y = 3x or even y - 3x = 0.
Use the chain rule to differentiate this equation and then solve for the unknown rate, leaving your answer in equation form.
Substitute the given information into this equation and find the unknown rate. Express your answer in the form dx/dt or dy/dt = a number.
We are trying to find the rate at which the distance between the cars is increasing, which is denoted by dz/dt.
The equation relating x and y can be derived from the Pythagorean theorem: x^2 + y^2 = z^2
Differentiate both sides of this equation with respect to t (time) using the chain rule:
2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)
Now we need to solve for dz/dt:
dz/dt = (x*dx/dt + y*dy/dt) / z
Substituting the given information:
x = 60 mph * 2 hours = 120 miles
y = 25 mph * 2 hours = 50 miles
z = distance between the cars
Substituting these values into the equation:
dz/dt = (120 mph * dx/dt + 50 mph * dy/dt) / z
Since dx/dt = -60 mph (negative because it is traveling south) and dy/dt = -25 mph (negative because it is traveling west):
dz/dt = (120 mph * (-60 mph) + 50 mph * (-25 mph)) / z
Simplifying:
dz/dt = (-7200 - 1250) mph^2 / z
dz/dt = (-8450) mph^2 / z
Therefore, the rate at which the distance between the cars is increasing 2 hours later is -8450 mph^2 divided by the current distance between the cars.
The two rates given are the speeds at which the cars are traveling. The south traveling car is moving at 60 mph, so its rate is given by dx/dt = 60 mph. The west traveling car is moving at 25 mph, so its rate is given by dy/dt = 25 mph.
We are trying to find the rate at which the distance between the cars is increasing, which is dz/dt.
To write an equation relating x and y, we can use the Pythagorean theorem, which states that the square of the hypotenuse (z) is equal to the sum of the squares of the other two sides (x and y). Therefore, we have the equation x^2 + y^2 = z^2.
Differentiating this equation using the chain rule, we have 2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt).
Since we are given the values of dx/dt (60 mph) and dy/dt (25 mph), we can substitute these values into the equation.
2(60) + 2(25) = 2z(dz/dt).
Simplifying this equation gives us 120 + 50 = 2z(dz/dt).
Substituting the values, we have 170 = 2z(dz/dt).
To find dz/dt, we can solve for it by dividing both sides of the equation by 2z.
dz/dt = 170 / (2z).
However, we need to find the value of z when the given time is 2 hours later.
Using the information that dx/dt = 60 mph and the time is 2 hours, we can calculate x = 60 mph * 2 hours = 120 miles.
Using the information that dy/dt = 25 mph and the time is 2 hours, we can calculate y = 25 mph * 2 hours = 50 miles.
Using these values of x and y, we can substitute them into our equation to find z.
120^2 + 50^2 = z^2.
Simplifying this equation gives us 14400 + 2500 = z^2.
Therefore, z^2 = 16900.
Taking the square root of both sides gives us z = √16900 = 130 miles.
Now we can substitute the value of z into the equation dz/dt = 170 / (2z).
dz/dt = 170 / (2 * 130).
Simplifying this equation gives us dz/dt = 0.65 mph.
Therefore, the rate at which the distance between the cars is increasing 2 hours later is 0.65 mph.
x = 60t
y = 25t
z = √(x^2+y^2) = 65t
All that chain rule stuff is irrelevant here, since dz/dt = a constant.