Triangle tangent to circle problem using derivatives

In summary, the problem is about a metal bar attached to a circle and a point that can slide along the x-axis. The goal is to find x as a function of the angle theta, and the angular speed at two specific values of theta. This can be done using the law of cosines.
  • #1
oates151
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Homework Statement

Tangent.jpg


A metal bar of length l in the figure below has one end attached at a point P to a circle ofradius a < l. Point Q at the other end can slide back and forth along the x–axis.

(a) Find x as a function of θ (θ=angle POQ).
(b) Assume the lengths are in centimeters and the angular speed,dθ/dt, is 2 radians persecond counter clockwise. Find the speed at which point Q is moving when θ =π/2 and when θ =π/4. Give units.



Homework Equations





The Attempt at a Solution



Having a hard time understanding what to do for this problem.
 
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  • #2
'l' is fixed and 'a' is fixed. theta is variable. If theta=0 then x=l+a, if theta=(-pi) then x=l-a, right? You want to express x as function of theta. Use some trig, like the law of cosines.
 
  • #3
Can you find x in terms of ℓ, a, and θ .

Use the law of cosines.
 

1. What is the relationship between a triangle and a circle in this problem?

In this problem, the triangle is tangent to the circle, meaning that it touches the circle at exactly one point. This creates a right angle between the triangle and the circle.

2. How can derivatives be used to solve this problem?

Derivatives can be used to find the slope of the tangent line at the point where the triangle and circle touch. This slope can then be used to find the angle between the tangent line and the x-axis, which can be used to find the angles of the triangle.

3. What information is needed to solve this problem using derivatives?

To solve this problem using derivatives, you will need the radius of the circle, the coordinates of the point where the triangle and circle touch, and the equation of the circle.

4. Can this problem be solved without using derivatives?

Yes, this problem can also be solved using basic geometry and trigonometry principles. However, using derivatives can provide a more efficient and precise solution.

5. What are some real-life applications of this problem?

This problem has many real-life applications, such as in engineering and architecture. For example, it can be used to calculate the slope of roads or the angle of support beams in a building that are tangent to a circular structure.

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