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Math Help - Related Rates & Angle of Elevation Help??

  1. #1
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    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.

    Again, the answer is 0.125 rad/s
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Related Rates & Angle of Elevation Help??

    1.) At time t, we may state:

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

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

    We are given the conditions:

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

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

    So, use these to find the value of \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) \tan(\theta)=\frac{y}{x}

    We also know by Pythagoras that:

    (2) x^2+y^2=5^2

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

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

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