   Chapter 2.8, Problem 8E

Chapter
Section
Textbook Problem

# The area of a triangle with sides of lengths a and b and contained angle θ is A = 1 2 a b   sin θ (a) If a = 2 cm, b = 3 cm, and 0 increases at a rate of 0.2 rad/ min, how fast is the area increasing when θ = π 3 ?(b) If a = 2 cm, 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 = π 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 = 2 cm, b = 3 cm, and θ = π 3 ?

To determine

a)

To find:

How fast the area is increasing when θ=π3

Explanation

1. Formula used:

i. Area of triangle A=12 absinθ

ii.

ddx(sinx)=cosx

iii. Chain rule of differentiation:

ddtfx=ddxfx*dxdt

2. Given:

a = 2 cm, b = 3 cm,dθdt=0.2 rad/min

3. Calculations:

Consider A=12 absinθ

Differentiate both sides with respect to t

To determine

b)

To find:

How fast the area is increasing when b =3 and θ=π3

To determine

c)

To find:

How fast the area is increasing when a= 2, b = 3 and θ=π3

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