Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is
A = (1/2) ab sinθ.


a. How is dA/dt related to dθ/dt if a and b are constant?

Verified step by step guidance

1. Start by understanding the formula for the area of a triangle: A = (1/2)ab sin(θ). Here, a and b are constants, and θ is the variable that changes over time.
2. To find how dA/dt is related to dθ/dt, apply the chain rule of differentiation. Since a and b are constants, differentiate A with respect to θ: dA/dθ = (1/2)ab cos(θ).
3. Now, use the chain rule to relate dA/dt to dθ/dt. The chain rule states that dA/dt = (dA/dθ) * (dθ/dt).
4. Substitute the expression for dA/dθ from step 2 into the chain rule formula: dA/dt = (1/2)ab cos(θ) * dθ/dt.
5. This equation shows that the rate of change of the area with respect to time, dA/dt, is directly proportional to the rate of change of the angle with respect to time, dθ/dt, and depends on the cosine of the angle θ.

Key Concepts

Derivative

The derivative represents the rate of change of a function with respect to a variable. In this context, dA/dt is the derivative of the area A with respect to time t, indicating how the area changes as the angle θ changes over time.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It allows us to find the derivative of a function with respect to an intermediate variable, such as θ, when the function is expressed in terms of another variable, like t. Here, it helps relate dA/dt to dθ/dt.

Trigonometric Functions

Trigonometric functions, such as sine, are essential in relating angles to side lengths in triangles. The function sinθ is used in the formula for the area of a triangle, and understanding its derivative is crucial for determining how changes in θ affect the area A when differentiating A = (1/2) ab sinθ.