Two fixed sides of a triangle are 176 cm and 16 cm in length, and the angle between them is increasing at a rate of 0.02 radians per second. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is [tex]\frac{\pi}{3}[/tex].

By utilizing the area formula for a triangle and the given values, we calculate the derivative of the area with respect to time to find the rate at which the area of the triangle is increasing.

Explanation:

First, let's clarify the problem. We have a triangle with two fixed sides with lengths of 176 cm and 16 cm. The angle between these two fixed sides is increasing at a rate of 0.02 radians per second. We need to find the rate at which the area of the triangle is increasing when the angle between the fixed sides is π/3.

To start, we can use the area formula for a triangle which states that the area (A) of a triangle is 1/2 * base * height. In this case, the base and height are our fixed sides with lengths 176 cm and 16 cm. So, the area of the triangle becomes A = 1/2 * 176 cm * 16 cm * sinθ, where θ is the angle between the two sides.

Next, we take the derivative of the area with respect to time (t). This will give us dA/dt = 1/2 * 176 cm * 16 cm * cosθ * dθ/dt. Given that dθ/dt = 0.02 rad/s, we can plug this value, along with θ = π/3, into the equation to find the rate at which the area of the triangle is changing.

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