Let’s work through the problem step by step. --- ### **Given:** - Two planes converge toward the same point at right angles. - At a certain instant: - Plane 1: 225 miles from the point, moving at 450 mph. - Plane 2: 300 miles from the point, moving at 600 mph. Let: - \( x \) = distance from Plane 1 to the intersection (225 miles, decreasing at 450 mph) - \( y \) = distance from Plane 2 to the intersection (300 miles, decreasing at 600 mph) - \( s \) = distance between the two planes We are asked: (a) **At what rate is the distance between the planes decreasing at this instant?** --- ## **Step 1: Relate the variables** The planes are approaching the intersection at right angles, so by the Pythagorean theorem: \[ s^2 = x^2 + y^2 \] --- ## **Step 2: Differentiate with respect to time \( t \)** \[ 2s \frac{ds}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} \] Divide both sides by \( 2 \): \[ s \frac{ds}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt} \] \[ \frac{ds}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{s} \] --- ## **Step 3: Plug in known values** At this instant: - \( x = 225 \) miles, \( \frac{dx}{dt} = -450 \) mph (distance is **decreasing**) - \( y = 300 \) miles, \( \frac{dy}{dt} = -600 \) mph (distance is **decreasing**) First, find \( s \): \[ s = \sqrt{x^2 + y^2} = \sqrt{225^2 + 300^2} = \sqrt{50625 + 90000} = \sqrt{140625} = 375 \text{ miles} \] --- ## **Step 4: Calculate \( \frac{ds}{dt} \)** Plug all values in: \[ \frac{ds}{dt} = \frac{225 \cdot (-450) + 300 \cdot (-600)}{375} \] \[ = \frac{-101250 - 180000}{375} \] \[ = \frac{-281250}{375} \] \[ = -750 \text{ mph} \] --- ### **Final Answer for (a):** \[ \boxed{750 \text{ miles per hour}} \] (The negative sign means the distance is **decreasing** at this rate.) --- ## **Step 5: Time until the planes meet (for part b)** Find the time until both planes reach the intersection (when both \( x \) and \( y \) reach zero): \[ t_x = \frac{x}{|\frac{dx}{dt}|} = \frac{225}{450} = 0.5 \text{ hours} \] \[ t_y = \frac{y}{|\frac{dy}{dt}|} = \frac{300}{600} = 0.5 \text{ hours} \] Both planes will reach the intersection in **0.5 hours (30 minutes)**. --- ### **Final Answer for (b):** \[ \boxed{0.5 \text{ hours (or 30 minutes)}} \] --- ## **Summary of Answers:** **(a)** The distance between the planes is decreasing at a rate of **750 mph**. **(b)** The air traffic controller has **0.5 hours (30 minutes)** to act.

Let's work through the problem carefully, detailing each step and calculation. --- ### **Problem Recap:** - Two planes are at the same altitude, approaching a common point. - At a particular instant: - Plane 1 is **225 miles** from the point, moving **toward** it at **450 mph**. - Plane 2 is **300 miles** from the point, moving **toward** it at **600 mph**. - The planes are at right angles to each other (perpendicular paths). - **Questions:** 1. How fast is the distance between the two planes decreasing at this instant? 2. How much time before they reach the point? --- ### **Part 1: Find the rate at which the distance between the planes is decreasing** --- ### **Step 1: Define variables** Let: - \( x(t) \): distance of Plane 1 from the intersection point (current value: 225 miles) - \( y(t) \): distance of Plane 2 from the intersection point (current value: 300 miles) - \( s(t) \): distance between the two planes at time \( t \) Because the planes are at right angles: \[ s^2 = x^2 + y^2 \] --- ### **Step 2: Find the derivatives** Differentiate both sides with respect to time \( t \): \[ 2s \frac{ds}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} \] Divide through by 2: \[ s \frac{ds}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt} \] Solve for \( \frac{ds}{dt} \): \[ \boxed{ \frac{ds}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{s} } \] --- ### **Step 3: Plug in the known values** At the instant: - \( x = 225 \) miles - \( y = 300 \) miles - \( \frac{dx}{dt} = -450 \) mph (since the plane is approaching the point) - \( \frac{dy}{dt} = -600 \) mph Note: Negative because the distances are decreasing. Calculate \( s \): \[ s = \sqrt{225^2 + 300^2} = \sqrt{ (225)^2 + (300)^2 } \] Compute squares: \[ 225^2 = 50,625 \\ 300^2 = 90,000 \] Sum: \[ 50,625 + 90,000 = 140,625 \] Square root: \[ s = \sqrt{140,625} = 375 \text{ miles} \] --- ### **Step 4: Compute \( \frac{ds}{dt} \)** Now, substitute all known values: \[ \frac{ds}{dt} = \frac{(225)(-450) + (300)(-600)}{375} \] Calculate numerator: \[ 225 \times (-450) = -101,250 \\ 300 \times (-600) = -180,000 \] Sum numerator: \[ -101,250 + (-180,000) = -281,250 \] Divide by \( s = 375 \): \[ \frac{ds}{dt} = \frac{-281,250}{375} \] Perform division: \[ \frac{281,250}{375} = 750 \] So, \[ \frac{ds}{dt} = -750 \text{ mph} \] **Interpretation:** - The negative sign indicates the distance is decreasing. - The **rate at which the distance between the planes is decreasing** is **750 mph**. --- ### **Final answer for Part (a):** \[ \boxed{ \textbf{The distance between the planes is decreasing at } \boxed{750 \text{ mph}} } \] --- ### **Part 2: Time before the planes reach the intersection** --- ### **Step 5: Calculate the time for each plane to reach the intersection** Since both planes are approaching the point at constant speeds: - For Plane 1: \[ t_{x} = \frac{\text{distance}}{\text{speed}} = \frac{225}{450} = 0.5 \text{ hours} \] - For Plane 2: \[ t_{y} = \frac{300}{600} = 0.5 \text{ hours} \] **Interpretation:** - Both planes will reach the intersection in **0.5 hours** (30 minutes). - They are expected to arrive simultaneously, assuming constant speeds. --- ### **Final answer for Part (b):** \[ \boxed{ \textbf{The planes will reach the intersection in } \boxed{0.5 \text{ hours}} \text{ (30 minutes)} } \] --- ## **Summary:** | Part | Answer | Explanation | |---------|------------------------------|--------------------------------------------------------------| | (a) | **750 mph** (decreasing rate) | The rate at which the distance between the planes is decreasing at that instant. | | (b) | **0.5 hours (30 minutes)** | Time remaining before both planes reach the intersection point. | --- If you need further clarification or additional calculations, feel free to ask!

(a) The distance between the planes is decreasing at **750 miles per hour**. (b) The planes will reach the intersection in **0.5 hours (30 minutes)**.