Related Rate -finding the Rate of Change of an Angle

In summary: Well if you know geometry, you will notice that the angle can be given as:\tan{\theta}=\frac{x}{2}\implies\theta=\arctan{\frac{x}{2}}Use the info I posted above when taking this derivative (I assume you know the derivative of arctan(x)).Then, use the chain rule:dθ/dt= [1/(1+ x^2/4)] dx/dtThis can be simplified to:dθ/dt= -[1/(1+ x^2)]
  • #1
jaggtagg7
6
0
here is the problem i was trying to do:

A baseball player stands 2 feet from home plate and watches a pitch fly by. Find the rate D(theta)/dt at which his eyes must move to wach a fastball with dx/dt=-130 ft/s as it crosses homeplate at x=0.

now there is a nice diagram of a right trianlge with x labled as the distance from the ball from the plate and theta as the angle from the player's eyes to the ball.

where I'm confused is how exactly i relate these. not sure what trig function to use, or then how to solve it.

also what would be some general rules to follow when solving any related rate problem involving an angle?

PS: this is not a homework problem, but rather on i was trying to solve for fun, so I'm not really interested in the answer but more of how you solve it.
 
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  • #2
jaggtagg7 said:
here is the problem i was trying to do:

A baseball player stands 2 feet from home plate and watches a pitch fly by. Find the rate D(theta)/dt at which his eyes must move to wach a fastball with dx/dt=-130 ft/s as it crosses homeplate at x=0.

now there is a nice diagram of a right trianlge with x labled as the distance from the ball from the plate and theta as the angle from the player's eyes to the ball.

where I'm confused is how exactly i relate these. not sure what trig function to use, or then how to solve it.

also what would be some general rules to follow when solving any related rate problem involving an angle?

PS: this is not a homework problem, but rather on i was trying to solve for fun, so I'm not really interested in the answer but more of how you solve it.
Please scan in the diagram if you can.

I'd have to look at it, but generally what you do is this:

Set up θ in terms of x. Then you can differentiate and notice that dθ/dx = (dθ/dt) / (dx/dt), which you can solve for dθ/dt.

Alex
 
Last edited:
  • #4
jaggtagg7 said:
http://img.photobucket.com/albums/v260/nendalauka/29.jpg

theres a link to a scan of the entire problem + its diagram.

thanks
Well if you know geometry, you will notice that the angle can be given as:

[tex]\tan{\theta}=\frac{x}{2}\implies\theta=\arctan{\frac{x}{2}}[/tex]

Use the info I posted above when taking this derivative (I assume you know the derivative of arctan(x)).

Alex
 
  • #5
ok, gotcha thus far. but now what i am unsure of is how you would implicity take the derivative for arctan(x/2). i can't recall how u would treat the (1/2)x in the derivative.

something like this?

dθ/dt = [1/(1+ x^2/4)] dx/dt

that can't be right, because when i substituted, i ened up with dθ/dt= 1/1 *-130
... :/

thanks for your help.
 
  • #6
jaggtagg7 said:
ok, gotcha thus far. but now what i am unsure of is how you would implicity take the derivative for arctan(x/2). i can't recall how u would treat the (1/2)x in the derivative.

something like this?

dθ/dt = [1/(1+ x^2/4)] dx/dt

that can't be right, because when i substituted, i ened up with dθ/dt= 1/1 *-130
... :/

thanks for your help.
Chain rule! You must also multiply by the derivative of x/2.

Alex
 

What is the concept of related rates?

The concept of related rates involves finding the rate of change of one variable with respect to another variable. This is often used in mathematics and physics to solve problems involving changing quantities.

How do you find the rate of change of an angle?

To find the rate of change of an angle, one must differentiate the angle with respect to time. This means taking the derivative of the angle equation with respect to time, and plugging in the given values to solve for the rate of change.

What are some real-life applications of related rates?

Related rates can be applied to many real-life situations, such as calculating the speed of a moving object, determining the rate of water flowing into a tank, or finding the rate at which the shadow of an object is changing.

What is the relationship between related rates and the chain rule?

The chain rule is a fundamental rule in calculus that is used to find the derivative of a composite function. It is also used in related rates problems, where the rate of change of one variable is dependent on the rate of change of another variable.

What are some common mistakes when solving related rates problems?

Some common mistakes when solving related rates problems include not correctly identifying the variables and their rates of change, not setting up the problem correctly, and not using the correct formula or equation. It is important to carefully read the problem and understand the relationship between the variables before attempting to solve.

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