These are CALC 1 question, please solve it with details, Thank you Edges of a cu

Question

These are CALC 1 question, please solve it with details, Thank you

Edges of a cube are expanding at a rate of 5 cm/sec. how fast is the volume of the cube changing when each edge is 3 cm long. An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 225 miles from the point, moving at 450 mi/h. the other plane is 300 mi from the point moving at 600 mi/h. a) At what rate is the distance between the planes decreasing? b) How much time does the air traffic controller have to get one of the planes on a different flight path?

Explanation / Answer

a) x = 225, dx/dt = 450

y = 300 , dy/dt = 600

From pythagorean's theorem x2 + y2 = z2

2252 + 3002 = z2

z = 375

Take the derivative to find dz/dt, x2 + y2 = z2

2 x dx/dt + 2 y dy/dt = 2 z dz/dt

x [dx/dt] + y [dy/dt] = z dz/ [dt]

( 225 * 450) + (300 * 600) = 375 (dz/dt)

dz/dt = 750

b) Time = distance / rate

Time = 300 miles / 750 mi/hr

Time = 0.4 of a hour = 24 min

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