Monday

September 22, 2014

September 22, 2014

Posted by **Amy** on Thursday, May 5, 2011 at 4:54am.

2.) An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 mph. Find the rate at which the angle of elevation is changing when the angle is at 60 degrees.

- Calculus-rates -
**Reiny**, Thursday, May 5, 2011 at 7:43amMake a diagram showing the lamppost and the man's position any time.

Join the top of the lamppost to the man's top and extend it to the ground.

let the distance between the lamppost and the man along the ground be x ft, let the length of his shadow on the ground be y ft.

Since we have similar triangles, use the ratio

(x+y)/15 = y/6

6x + 6y = 15y

6x = 9y

2x = 3y

then 2dx/dt = 3dy/dt

2(5) = 3dy/dt

dy/dt = 10/3

So it appears that the position of the man is irrelevant and the shadow is changing at 10/3 ft/sec

Had it asked "how fast is his shadow moving ?" you would have added the 5ft/sec to the above answer.

2nd problem:

Again, make a diagram showing the plane moving in a line parallel to the ground.

Let the distance of the plane's path be x miles and the angle of elevation be Ø

I see that

tan Ø = 5/x

x tanØ = 5

x sec^2 Ø dØ/dt + tanØ dx/dt = 0

so when Ø = 60° , x = 5/√3

(5/√3)(1/3) dØ/dt + √3(600) = 0

dØ/dt = -1080 radians/hour

or 3/10 radians/sec , (I divided by 3600 to get to seconds)

(since derivatives in trig only work with radians, the answer obtained would be in radians)

**Answer this Question**

**Related Questions**

Calculus- rates - A man 6ft. tall walks at the rate of 5 ft/sec toward a ...

Calculus - Shadow Length A man 6 feet tall walks at a rate of 3 ft per second ...

calculus - My problem is that a stft above the sidewalk. a man 6ft tall walks ...

calculus - A woman 5.5 ft tall walks at a rate of 6 ft/sec toward a streetlight ...

Math - a man 6ft tall walks at the rate of 5ft/sec toward a streetlight that is ...

Calculus ~ Related Rates - A man 2 meters tall walks at the rate of 2 meters per...

Calculus - A street light is hung 18 ft. above street level. A 6-foot tall man ...

Calculus - A man 5.50 ft tall approaches a street light 18.0 ft above the ground...

Calculus - A woman 5 ft tall walks at the rate of 5.5 ft/sec away from a ...

diffirential calculus - a woman 6ft tall walks away from a light 10ft above the ...