Calculus: Related Rate Problem

Question

A sand bag is dropped from a balloon at a height of 60 meter when the angle of elevation to the sun is 30^0. The position of the sandbag is s(t)= 60-4.9t^2. Find the rate at which the shadow of the sand bag is travelling along the ground when the sand bad is at a height of 35 meters.

                                      

                                  60 meter

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                              /           |       

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                     /                     | 35 meter                   

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       /__________________  |

 

Answer 1

 

Hi Parveen,


According to the trigonometry of a triangle,

the shadow length, l(t), on the ground can be obtained as

     l(t) = s(t)/tan(30^º)

          = √3(60 - 4.9t²)

 

And, the shadow's travelling rate on the ground

is given by the first derivative of l(t),

     l'(t) = -9.8√3t

 

The time when the bag was 35 meters above the ground

can be obtained by solving

     60 - 4.9t² = 35

     t = 5√10/7

 

Plugging t = 5√10/7 into l'(t),

     l'(5√10/7) = -9.8√3(5√10/7)

                     = -7√30

                     = -38.34 [m/s]