# Thread: calculate the rate of change of this hot air balloon

1. ## calculate the rate of change of this hot air balloon

A hot air balloon rises straight up from a level field and is detected by a range finder 200 m from the lift off point. At that point, the angle of elevation of the balloon is 60 degrees, and the angle is increasing at the rate of 0.1 rad/min. Find the rate at which the balloon is rising when the angle of elevation of the balloon from the range finder is 45 degrees.

Could someone please show me the full steps

2. Originally Posted by differentiate
a hot air balloon rises straight up from a level field and is detected by a range finder 200 m from the lift off point. At that point, the angle of elevation of the balloon is 60 degrees, and the angle is increasing at the rate of 0.1 rad/min. Find the rate at which the balloon is rising when the angle of elevation of the balloon from the range finder is 45 degrees.

Could someone please show me the full steps

Code:
|.
|   .
|      .
|         .
|            .
|               .
|h                 .
|                     .
|                        .
|

img.top {vertical-align:15%;}

$\theta$   .
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
200
$\tan{\theta} = \frac{h}{200}$

$\sec^2{\theta} \cdot \frac{d\theta}{dt} = \frac{1}{200} \cdot \frac{dh}{dt}$

When the angle is 45 degrees:

$200 \sec^2{\left(\frac{\pi}{4}\right)} \frac{d\theta}{dt}= \frac{dh}{dt}$

$400\frac{d\theta}{dt}= \frac{dh}{dt}$

Now use the remaining information for when the angle is 60 degrees to finish the problem with simultaneous equations.

Mathemagister

3. sorry. but I still don't get it.

4. I put in theta = pi/3, with the remaining info and I got
dh/dt = 80 metres/min when theta = pi/3 and d(theta)/dt = 0.1

but how do i solve simultaneously with this info and the info,
400 (d(theta)/dt = dh/dt) ?