Two sides of a triangle are 4 m and 8 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is 𝜋3 rad.
Question
Two sides of a triangle are 4 m and 8 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is 𝜋3 rad.
Solution
The area of a triangle with two sides of lengths a and b, with an included angle θ, can be calculated using the formula:
Area = 0.5 * a * b * sin(θ)
Given that a = 4m, b = 8m, and dθ/dt = 0.06 rad/s, we want to find d(Area)/dt when θ = π/3 rad.
First, differentiate the area with respect to time (t), using the chain rule:
d(Area)/dt = 0.5 * a * b * cos(θ) * dθ/dt
Substitute the given values into the equation:
d(Area)/dt = 0.5 * 4m * 8m * cos(π/3) * 0.06 rad/s = 0.5 * 4m * 8m * 0.5 * 0.06 rad/s = 0.24 m²/s
So, the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is π/3 rad is 0.24 m²/s.
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