A plane flying with a constant speed of 5 km/min passes over a ground radar station at an altitude of 4 km and climbs at an angle of 35 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing 10 minutes later?
I got it down to [250-20cos(125)]/[sqrt(40000-400cos(125))] but it's wrong.
I would call the time after passing over the station as t minutes
then by cosine law
D^2 = 4^2 + (5t)^2 - (2)(4)(5t)cos125
= 25t^2 * (40cos125)t + 16
I usually don't take the square root, but rather do it implicitly, so
2D(dD/dt) = 50t + 40cos125
dD/dt = (25t + 20cos125)/D
when t=10
D^2 = 2500 - 400cos125 + 16
D = 47.818
and dD/dt = (250 + 20cos125)/47.818
= 4.988 km/min
Not right for some reason.
re-checked my calculations, found a few typing and calculator errors. The old brain just doesn't work properly after midnight
NEW VERSION:
I would call the time after passing over the station as t minutes
then by cosine law
D^2 = 4^2 + (5t)^2 - (2)(4)(5t)cos125
= 25t^2 - (40cos125)t + 16
I usually don't take the square root, but rather do it implicitly, so
2D(dD/dt) = 50t - 40cos125
dD/dt = (25t - 20cos125)/D
when t=10
D^2 = 2500 - 400cos125 + 16
D = 47.818
and dD/dt = (250 - 20cos125)/52.3969
= 4.99 km/min
check my calculations
Ahh, there we go. Can't believe such a little number could cause it to be wrong. Thanks.
To solve this problem, we can use trigonometry and the concept of rates. Let's break it down step by step:
1. Label the relevant quantities:
- Distance from the plane to the radar station: Let's call it d (in km).
- Altitude of the plane: 4 km.
- Angle of climb: 35 degrees.
- Speed of the plane: 5 km/min.
- Time: 10 minutes later.
2. Find the horizontal and vertical components of the plane's motion:
- The horizontal component is given by cos(35) * speed = cos(35) * 5.
- The vertical component is given by sin(35) * speed = sin(35) * 5.
3. Use the distance formula to relate the variables:
- In a right triangle, the hypotenuse (d) can be calculated using the Pythagorean theorem:
d^2 = (horizontal component)^2 + (vertical component + altitude)^2.
4. Differentiate the equation with respect to time (t):
- Take the derivative of both sides of the equation implicitly with respect to t:
2d * dd/dt = 2(horizontal component) * (d(horizontal component)/dt) + 2(vertical component + altitude) * (d(vertical component + altitude)/dt).
5. Simplify the equation and find the rate of change of distance, dd/dt:
- Substitute the given values: horizontal component = cos(35) * 5, vertical component = sin(35) * 5, altitude = 4.
- Substitute the rate of change of altitude (d(vertical component + altitude)/dt) = 0 (since it is not changing).
- Solve for dd/dt by substituting t = 10 minutes (remember to convert it to km/min).
By following these steps, you should be able to calculate the correct answer.