To solve the problem, we need to establish the relationship between the angle and the value of x in the context of the triangle. Let’s denote the angle as θ. As per the information provided, θ increases at a constant rate, specifically at 3 radians per minute.
Given the triangle, we can use trigonometric relationships to express x in terms of θ. Assuming x relates to the sine function:
- Let’s say x = r * sin(θ), where r is a constant length related to the triangle.
Now, we differentiate x with respect to time (t) to find dx/dt:
- Using the chain rule, we have:
- dx/dt = r * cos(θ) * dθ/dt
From above, we know dθ/dt = 3 radians/min.
Next, we need to evaluate this expression when x = 3 units:
- From the relation x = r * sin(θ), we can find the required angle θ:
- If we assume r remains a constant length (for example, let’s assume r = 3 for simplicity), we thus have sin(θ) = 3/r. If r = 3, sin(θ) = 1 implies θ = π/2 radians.
Using θ = π/2 in our differentiation:
- cos(θ) = cos(π/2) = 0.
This means:**
- dx/dt = r * cos(θ) * dθ/dt = 3 * 0 * 3 = 0 units/min.
In conclusion, at the moment when x is at 3 units, the rate at which x is increasing is 0 units per minute. This suggests that while the angle is changing, the corresponding x value remains constant at that point in the triangle. Looking at it from a geometric perspective, the triangle has reached its maximum attainable x value in the specific configuration with respect to the angle θ.