We are dealing with a related rates problem involving two cars moving away from a common starting point. One car travels north, and the other travels east. The goal is to determine how fast the distance between the two cars is changing at a specific moment.

Step 1: Define Variables

• Let $x$ represent the distance the eastbound car is from the starting point.
• Let $y$ represent the distance the northbound car is from the starting point.
• Let $z$ represent the distance between the two cars (the hypotenuse of the right triangle formed by their paths).
• We need to find $\frac{dz}{dt}$, the rate at which the distance between the cars is changing.

We are given:

• After 3 hours, the first car (northbound) is 60 miles away, traveling at 35 mph. Thus, $y(3) = 60$ miles and $\frac{dy}{dt} = 35$ mph.
• After 3 hours, the second car (eastbound) is 80 miles away, traveling at 65 mph. Thus, $x(3) = 80$ miles and $\frac{dx}{dt} = 65$ mph.

Step 2: Use the Pythagorean Theorem

The distance between the cars, $z$, is given by: $z^2 = x^2 + y^2$ Differentiating both sides of this equation with respect to time $t$ using implicit differentiation: $2z \frac{dz}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$ Simplifying: $z \frac{dz}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt}$ We can solve for $\frac{dz}{dt}$: $\frac{dz}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{z}$

Step 3: Calculate $z$ at $t = 3$ hours

We know:

• $x = 80$ miles
• $y = 60$ miles

Using the Pythagorean theorem: $z = \sqrt{x^2 + y^2} = \sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100 \text{ miles}$

Step 4: Plug in Values

Now, substitute the known values into the equation for $\frac{dz}{dt}$: $\frac{dz}{dt} = \frac{80(65) + 60(35)}{100}$ $\frac{dz}{dt} = \frac{5200 + 2100}{100} = \frac{7300}{100} = 73 \text{ mph}$

Final Answer:

The rate at which the distance between the two cars is changing after 3 hours is $73$ mph.

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Would you like further details or have any questions about this solution?

Here are 5 related questions to expand your understanding:

1. What if the second car traveled northwest instead of east?
2. How would the result change if both cars traveled at constant speeds but along non-perpendicular paths?
3. How can you determine the exact position of each car after 4 hours?
4. What is the significance of the Pythagorean theorem in related rates problems?
5. What if the speeds of the cars increased after 3 hours? How would that affect the rate of change?

Tip: When dealing with related rates, always remember to carefully differentiate all variables with respect to time using implicit differentiation.