Shadow Height Rate in Related Rates Problem

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In summary, the problem involves finding the rate at which the shadow of a sandbag dropped from a balloon at a height of 60m is moving when the sandbag is at a height of 35m. The position of the sandbag is given by s(t)=60-4.9t^2 and the shadow's path can be related to it using trigonometry. By setting the vertical leg (s(t)) equal to 35 and solving for t, the vertical rate can be found. This rate is then related to the horizontal rate by (vertical rate)=(horizontal rate)*tan(30).
  • #1
physicsman2
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Homework Statement


A sandbag is dropped from a balloon at a height of 60m when the angle of elevation to the sun is 30 degrees. Find the rate at which the shadow is at a height of 35 meters


Homework Equations


the position of the sandbag is given by s(t)=60-4.9t^2


The Attempt at a Solution


i tried to find the relation between the shadow's path and the position of the sandbag, but my efforts were futile.
 
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  • #2
The point below the sandbag, the position of the sandbag and the position of the shadow make a right triangle. Use trig.
 
  • #3
i know that but i need help trying to solve it
i don't know how to relate the shadow to something else
 
  • #4
One leg of your triangle has length s(t). You want to figure out the length of the other leg. One of the angles in your triangle is 30 degrees.
 
  • #5
i got the hypotenuse to be 70 and b to be 61 since a would be 35 since the problem asks when the height is 35
is this right and if so, how would i go from there I am still having trouble
 
  • #6
What's the relation between the vertical leg (s(t)) and the horizontal leg? Which trig function would be good to use? One of them is the ratio between the two legs.
 
  • #7
im sorry if the problem doesn't make sense
s(t)=60-4.9t^2 is the instantaneous position of the sandbag in the air, not a leg.
I think that's what makes this problem so confusing
so to find the sides of the triangle, i did sin(30)/35= the hypotenuse and tan(30)/35=the base of the triangle since the problem says when the sandbag is 35 feet in the air, giving us the 35 to use.
 
  • #8
s(t) is the position of the sandbag at time t. It's also the length of the vertical leg at time t. If h(t) is the length of the horizontal leg at time t, then yes, s(t)/h(t)=tan(30 degrees). So s(t)=tan(30 degrees)*h(t). The rates are then related by d/dt(s(t))=d/dt(h(t))*tan(30 degrees). The vertical rate is equal to the horizontal rate times tan(30 degrees). You need to find the vertical rate when s(t)=35 and then solve for the horizontal rate. The other lengths aren't important.
 
  • #9
okay so would 60-4.9t^2=xtan(30) where x=horizontal length so if you differentiate that you would get -9.8t=(dx/dt)tan(30)+xsec^2(30)
is this right and if so where would i go from here
I am sorry for being a pain but i really don't understand this problem
 
  • #10
Don't worry about being a pain, but no, you don't understand the problem. But you are getting closer. You have to figure out what the rate -9.8*t is at the point where 60-4.9*t^2=35. That means you have to solve the latter equation for t and plug it into the first equation to get the vertical rate. Once you have that, (vertical rate)=(horizontal rate)*tan(30). You don't have to differentiate the trig function, it's a constant!
 

1. How do you identify a problem as a related rates (shadow) problem?

A related rates (shadow) problem involves finding the rate of change of one quantity in relation to the rate of change of another related quantity. The key to identifying this type of problem is to look for a situation where two or more quantities are changing with respect to time.

2. What are the steps to solving a related rates (shadow) problem?

The steps to solving a related rates (shadow) problem are:

  1. Identify the relevant variables and their rates of change.
  2. Write an equation that relates the variables.
  3. Differentiate the equation with respect to time.
  4. Substitute in the given rates of change.
  5. Solve for the desired rate of change.

3. How do you handle units in related rates (shadow) problems?

When solving related rates (shadow) problems, it is important to keep track of the units for each variable and to make sure they are consistent throughout the problem. This will ensure that the final answer has the correct units.

4. Can you give an example of a related rates (shadow) problem?

An example of a related rates (shadow) problem is a person standing next to a streetlight. As they walk away from the light, their shadow increases in length at a certain rate. The problem may ask to find the rate at which the person is walking.

5. What are some common mistakes people make when solving related rates (shadow) problems?

Some common mistakes people make when solving related rates (shadow) problems include:

  • Forgetting to differentiate the equation with respect to time.
  • Using the wrong units or not keeping track of units throughout the problem.
  • Not clearly defining the variables and their rates of change.
  • Substituting in the wrong values for the rates of change.

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