Calculus 1

Question

Two sides of a triangle have lengths 12m and 15m. The angle between them in increasing at a rate of 2degrees/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60degrees?

Answer 1

Let the angle between the two sides be Θ, and r the length of the third side.

Then by the Law of Cosine,  r² = 15 ² + 12 ² - 2 (15)(12) cos Θ

Differentiate implicitly.

2r dr/dt = 360 sinΘ  dΘ/dt

dr/dt = (180 sinΘ dΘ/dt)/r

 

To find the required dr/dt, when dΘ/dt = 2 deg/min, and Θ =  60^o, you need to determine the value of r.

Use the law of cosine to solve for r.  ...... r² = 15 ² + 12 ² - 2 (15)(12) cos 60

Substitute all values into dr/dt = (180 sinΘ dΘ/dt)/r