The problem you provided involves two cars moving in perpendicular directions, and we need to determine how fast the distance between the two cars is increasing at a specific moment.

Given:

• Car 1 is moving south at 45 miles per hour.
• Car 2 is moving west at 60 miles per hour.
• When Car 1 is 3 miles from Math-town and Car 2 is 4 miles from Math-town, how fast is the distance between the two cars increasing?

Step 1: Define variables

• Let $x$ represent the distance of Car 2 (heading west) from Math-town. At the given moment, $x = 4$ miles.
• Let $y$ represent the distance of Car 1 (heading south) from Math-town. At the given moment, $y = 3$ miles.
• Let $z$ represent the distance between the two cars. We want to find $\frac{dz}{dt}$, the rate at which the distance between the two cars is increasing.

Step 2: Use the Pythagorean theorem

The two cars are moving in perpendicular directions, so their distances from Math-town form a right triangle with the distance between the two cars as the hypotenuse. The relationship between $x$, $y$, and $z$ is given by:

$z^2 = x^2 + y^2$

Step 3: Differentiate with respect to time

To find the rate at which the distance between the cars is increasing, we differentiate both sides of the equation with respect to time $t$:

$2z \frac{dz}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$

Simplifying:

$z \frac{dz}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt}$

Step 4: Plug in known values

At the given moment:

• $x = 4$ miles,
• $y = 3$ miles,
• $\frac{dx}{dt} = 60$ miles per hour (speed of Car 2),
• $\frac{dy}{dt} = 45$ miles per hour (speed of Car 1).

First, we calculate $z$, the distance between the two cars:

$z = \sqrt{x^2 + y^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ miles}.$

Now, substitute all the values into the differentiated equation:

$5 \frac{dz}{dt} = 4(60) + 3(45)$

$5 \frac{dz}{dt} = 240 + 135 = 375$

$\frac{dz}{dt} = \frac{375}{5} = 75 \text{ miles per hour}.$

Conclusion:

The distance between the two cars is increasing at a rate of 75 miles per hour when Car 1 is 3 miles and Car 2 is 4 miles from Math-town.

Would you like more details or have any other questions? Here are 5 related questions for deeper understanding:

1. What happens to the rate of distance change if Car 1 slows down to 30 miles per hour?
2. How would the scenario change if Car 2 was heading east instead of west?
3. Can you derive the equation if one of the cars was moving at a variable speed?
4. What would the total distance be after 1 hour of movement at the given speeds?
5. How would the solution change if the cars were not moving perpendicularly?

Tip:

When solving related rates problems, always start by defining a geometric relationship (like the Pythagorean theorem in this case) and then differentiate with respect to time.