SOLUTION: When a rocket is 2 kilometers high, its moving vertically upward at a speed of 300 kilometers per hour. At that instant, how fast is the angle of elevation of the rocket increasing

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Question 389724: When a rocket is 2 kilometers high, its moving vertically upward at a speed of 300 kilometers per hour. At that instant, how fast is the angle of elevation of the rocket increasing as seen by an observer on the ground 5 kilometers from the launching pad. State the units of measurement.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
If y is the height and a+ the angle of elevation, then:
dy%2Fdt+=+300+km%2Fhr ...(1)
y+=+5+%2Atan%28a%29....(2)

Differentiating (2) with respect to time:
dy%2Fdt+=+5%2A+sec%5E2%28a%29%28da%2Fdt%29

Substituting for dy%2Fdt from (1):

300+=+5+%2Asec%5E2%28a%29+%28da%2Fdt%29

da%2Fdt+=+%28300+%2F+5+sec%5E2%29a

= 60+cos%5E2%28a%29 ...(2)

When y+=+2km:
tan%28a%29+=+2%2F5
cos%5E2%28a%29
= 1+%2F+sec%5E2%28a%29
= 1+%2F+%281+%2B+%282%2F5%29%5E2%29%29
= 1+%2F+%28+1+%2B+%284%2F25%29%29
= 25+%2F+29
Substituting this value in (2):
da%2Fdt+=+60+%2A+%2825+%2F+29%29
= %281500+%2F+29+%29rad%2Fhr
= %281500+%2F+%2829+%2A+3600%29%29+%28rad%2Fsec%29
= 5+%2F+%2829+%2A+12%29
= %285+%2F+348%29+%28rad%2Fsec%29