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Question

The area if a triangle with sides of lengths a and b and contained angle theta is A=1/2 ab sin theta.

a) If a= 2 cm, b = 3cm, and theta increases at a rate of 0.2 rad/min, how fast is the area increasing when theta = pi/3?

 

b) If a = 2cm, b increases at a rate of 1.5 cm/min, and θ increases at a rate of 0.2 rad/min, how fast is the area increasing when b = 3 cm and Θ = pi/3?

 

c) If a increases at a rate of 2.5 cm/min, b increases at a rate of 1.5 cm/min, and θ increases at a rate of 0.2 rad/min, how fast is the area increasing when a = 2cm, b =3cm, and Θ = pi/3?

Answer 1

a= 2cm b=3cm
A = 1/2 ( abSin theta)

( a) dA/dt = d/dt (1/2 ab Sin theta)

=1/2 (ab) d/dt( Sin theta)
= (1/2 ab Cos theta) d/dt (theta)
= (1/2)x 2x3x Cos(pi/3) x 0.2
= 3x1/2x x0.2 cm2/min as Cos pi/3 =12
= 0.3 cm2 /min

(b) dA/dt = d/dt ( 1/2 ab Sin theta) = 1/2 a x d/dt ( bSin theta)
This time both b and theta are variables

dA/dt = 1/2 xax (bcos theta d/dt theta + d/dt (b) x Sin theta

= 1/2x2x 3x Cos pi/3 +1/2 x2x 1.5 x Sin pi/3 = 3x1/2 +1.5 x sqr (3)/2 cm2 /min

(c) in this case take all the three a, b and theta as variables and differentiate for three variables

We have used Sin pi/3 = sqr3/2 and Cos pi/3 =1/2