Let's solve the problem step by step: ### **Given:** - Plane 1 is 225 miles from the intersection moving at **450 miles/hour**. - Plane 2 is 300 miles from the intersection moving at **600 miles/hour**. - They are converging at right angles. Let: - \( x \) = distance of plane 1 from the intersection (initially 225 miles) - \( y \) = distance of plane 2 from the intersection (initially 300 miles) - \( s \) = distance between the planes Their rates: - \( \frac{dx}{dt} = -450 \) mi/hr (negative because \( x \) is **decreasing**) - \( \frac{dy}{dt} = -600 \) mi/hr (negative because \( y \) is **decreasing**) --- ## (a) **At what rate is the distance between the planes decreasing?** Use the Pythagorean Theorem: \[ s^2 = x^2 + y^2 \] Differentiate both sides with respect to \( t \): \[ 2s \frac{ds}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} \] \[ s \frac{ds}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt} \] \[ \frac{ds}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{s} \] ### Plug in the values at the instant: - \( x = 225 \) miles - \( y = 300 \) miles - \( \frac{dx}{dt} = -450 \) mi/hr - \( \frac{dy}{dt} = -600 \) mi/hr First, calculate \( s \): \[ s = \sqrt{x^2 + y^2} = \sqrt{225^2 + 300^2} = \sqrt{50625 + 90000} = \sqrt{140625} = 375 \text{ miles} \] Calculate numerator: \[ x \frac{dx}{dt} + y \frac{dy}{dt} = 225 \cdot (-450) + 300 \cdot (-600) = -101250 + (-180000) = -281250 \] Calculate \( \frac{ds}{dt} \): \[ \frac{ds}{dt} = \frac{-281250}{375} = -750 \text{ mi/hr} \] ### **Final Answer for (a):** \[ \boxed{750 \text{ miles per hour \ (the distance between the planes is decreasing at this rate)}} \] --- ## (b) **How much time does the air traffic controller have before the planes converge?** We need the time when **either \( x \) or \( y \) reaches 0** (i.e., the first one to reach the intersection). - For Plane 1: \( x = 225 \) miles, speed \( 450 \) mi/hr - For Plane 2: \( y = 300 \) miles, speed \( 600 \) mi/hr Time for each to reach the intersection: \[ t_1 = \frac{225}{450} = 0.5 \text{ hr} \] \[ t_2 = \frac{300}{600} = 0.5 \text{ hr} \] **Both will reach the intersection at the same time: 0.5 hr = 30 minutes.** ### **Final Answer for (b):** \[ \boxed{0.5 \text{ hours, or 30 minutes}} \] The controller has 30 minutes to get one of the planes on a different flight path.