Question

Two sides of a triangle are 7 m and 10 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate (in m²/s) at which the area of the triangle is increasing when the angle between the sides of fixed length is π/3 rad.

Answer 1

To find the rate at which the area of a triangle is increasing, we use the area formula for a triangle and differentiate with respect to time, incorporating the rate of change of the angle. The calculated rate of change of the area, when the angle is π/3 rad and the angle's rate of change is 0.06 rad/s, is 1.75 m²/s.

Step-by-step explanation:

The problem at hand requires us to find the rate at which the area of a triangle is increasing, given two sides of 7 m and 10 m, and the angle between them increasing at a rate of 0.06 rad/s. We know that the area of a triangle can be calculated using the formula A = 1/2 * a * b * sin(θ), where a and b are the sides of the triangle and θ is the angle between them.

When the angle θ is π/3 rad, we can calculate the rate of change of the area using the derivative of the area with respect to time, denoted as dA/dt. Since we're given dθ/dt, we can use the chain rule to find dA/dt = 1/2 * ab * cos(θ) * dθ/dt. Substituting the given values, dA/dt turns out to be 1.75 m²/s.