# Related Rates & Angle of Elevation Help??

• Jan 22nd 2013, 08:59 AM
iuulia338
Related Rates & Angle of Elevation Help??
I'm not sure how to do these two questions. Any help with both or one is appreciated!!

1) A balloon rises into the air started at point P. AN observer, 100m from P looks at the balloon and the angle theta between her line of sight and the ground increases at a rate of 1/20 rads per second. Find the velocity of the balloon where theta = pi/4

The answer is 10m/s I'm just not sure how to get it.

2) One end of a ladder of length 5m slides down a verticle wall. When the upper end of the ladder is 3m from the ground, it has a downward velocity of 0.5m/s. Find the rate at which the angle of elevation is changing at that time.

• Jan 22nd 2013, 09:55 AM
MarkFL
Re: Related Rates & Angle of Elevation Help??
1.) At time $\displaystyle t$, we may state:

$\displaystyle \tan(\theta)=\frac{h}{100}$

Implicitly differentiating with respect to $\displaystyle t$, and solve for $\displaystyle \frac{dh}{dt}$

We are given the conditions:

$\displaystyle \theta=\frac{\pi}{4}$

$\displaystyle \frac{d\theta}{dt}=\frac{1}{20}\,\frac{\text{rad}} {\text{s}}$

So, use these to find the value of $\displaystyle \frac{dh}{dt}$, which represents the vertical speed of the balloon at the given point.

2.) Let x be the distance of the base of the ladder from the wall and y be the height of the top of the ladder on the wall. We may then relate the angle of elevation to these variables with

(1) $\displaystyle \tan(\theta)=\frac{y}{x}$

We also know by Pythagoras that:

(2) $\displaystyle x^2+y^2=5^2$

Differentiate (1) with respect to $\displaystyle t$ and solve for $\displaystyle \frac{d\theta}{dt}$, and replace the resulting trig. function by its ratio definition.

Differentiate (2) with respect to $\displaystyle t$ to find $\displaystyle \frac{dx}{dt}$.

Now, you should have everything you need to answer the question.