1. **Problem Statement:** Two planes are flying at right angles to each other, both at the same altitude, converging on a point. One plane is 225 miles from the point, moving at 450 miles per hour. The other plane is 300 miles from the point, moving at 600 miles per hour. We want to find: (a) The rate at which the distance $s$ between the planes is decreasing. (b) The time the air traffic controller has to change one plane's flight path before they meet. 2. **Set up variables and formula:** Let $x$ be the distance of the first plane from the point, and $y$ be the distance of the second plane from the point. The distance between the planes is the hypotenuse $s$ of a right triangle with legs $x$ and $y$. By the Pythagorean theorem: $$s^2 = x^2 + y^2$$ 3. **Given data:** - $x = 225$ miles, moving towards the point at $\frac{dx}{dt} = -450$ mph (negative because distance is decreasing). - $y = 300$ miles, moving towards the point at $\frac{dy}{dt} = -600$ mph. 4. **Find $s$ at this instant:** $$s = \sqrt{x^2 + y^2} = \sqrt{225^2 + 300^2} = \sqrt{50625 + 90000} = \sqrt{140625} = 375 \text{ miles}$$ 5. **Differentiate the Pythagorean relation with respect to time $t$:** $$2s \frac{ds}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$$ Simplify by dividing both sides by 2: $$s \frac{ds}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt}$$ 6. **Solve for $\frac{ds}{dt}$:** $$\frac{ds}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{s}$$ 7. **Substitute known values:** $$\frac{ds}{dt} = \frac{225 \times (-450) + 300 \times (-600)}{375} = \frac{-101250 - 180000}{375} = \frac{-281250}{375} = -750 \text{ mph}$$ The negative sign indicates the distance between the planes is decreasing at 750 mph. 8. **Bonus: Time until planes meet (distance $s=0$):** Since $s$ is decreasing at 750 mph and currently $s=375$ miles, $$\text{time} = \frac{s}{|\frac{ds}{dt}|} = \frac{375}{750} = 0.5 \text{ hours} = 30 \text{ minutes}$$ **Final answers:** (a) The distance between the planes is decreasing at a rate of **750 miles per hour**. (b) The air traffic controller has **30 minutes** to change one plane's flight path before they meet.