Question

Two sides of a triangle are 2 meters and 3 meters and the angle between them is increasing at 0.5 radians/second when the angle is pi/4.

1. How fast is the distance between the tips increasing?

2. How fast is the area increasing?

I got part of #1 but I am getting multiple answers. Help me clear up the confusion. Thanks Mary!

Answer 1

I would assume you are using the cosine law.
let the distance between the two tips be x

x^2 = 4 + 9 -2(2)(3)cosØ
x^2 = 13 - 12cosØ
2x dx/dt = 12sinØ dØ/dt
given: dØ/dt = .5 , Ø = π/4

for Ø = π/4
x^2 = 13 - 12cosπ/4
= 4.5147..
x = appr 2.124..

dx/dt = 12 sinπ/4 (.5) /2.124..
= 1.9967
or appr 2 m/second

2. Assuming you want the rate of increase of the area for the given angle as well

area = (1/2)(2)(3)sinØ = 3sinØ
d(area) = 3cosØ dØ/dt
= 3(√2/2)(.5) = 3√2/4 m^2/second
= appr 1.06 m^2/s