Two sides of a triangle have lengths 11 m and 18 m...

Question

Two sides of a triangle have lengths 11 m and 18 m. The angle between them is increasing at a rate of 2°/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60°?

Answer 1

This is a very interesting problem! Can you recall any relation between two sides of a triangle and the angle between them?

 

You should remmber the Law of Cosines, which is:

 

c² = a² + b² - 2ab cos(θ)

 

Does that look familiar?

 

Here, c is the longest side, and a and b are the shorter sides. Theta is the angle between them.

 

Use implicit differentiation to get...

 

2c (dc/dt) = -2ab(-sin(θ)) dθ/dt

 

You want to solve for (dc/dt) now. Remember to convert your angles from degrees to radians. 60 degrees is π/3. 2 degrees/min converts to 2*π/180 = π/90

 

2c *dc/dt = 2*(11)(18)*sin(π/6)*π/90

              = 2(11)(18)(√3/2)(π/90)

              = 11*18*√3*π/90

 

Find 'c' using the original law of cosines, then solve for dc/dt. I will leave the final steps for you to figure out.

 

I hope this helps!