Final answer:
dl/dt = 10
Step-by-step explanation:
To find dl/dt, we need to differentiate the expression l = sqrt(x^2 * y^2) with respect to t. We can use the chain rule to do this. Let's start by differentiating with respect to x: dl/dx = (1/2) * (x^2 * y^2)^(-1/2) * 2x * y^2 = xy^2/sqrt(x^2 * y^2) = xy/sqrt(x^2). Next, let's differentiate with respect to y: dl/dy = (1/2) * (x^2 * y^2)^(-1/2) * x^2 * 2y = x^2y/sqrt(x^2 * y^2) = xy/sqrt(y^2). Finally, we can use the chain rule to find dl/dt: dl/dt = (dl/dx * dx/dt) + (dl/dy * dy/dt) = (xy/sqrt(x^2) * dx/dt) + (xy/sqrt(y^2) * dy/dt). Substituting the given values for dx/dt = -3, dy/dt = 4, x = 4, and y = 2, we can calculate dl/dt as follows:
- dl/dt = (xy/sqrt(x^2) * dx/dt) + (xy/sqrt(y^2) * dy/dt)
- dl/dt = (4*2/sqrt(4^2) * -3) + (4*2/sqrt(2^2) * 4)
- dl/dt = (8/4 * -3) + (8/2 * 4)
- dl/dt = -6 + 16
- dl/dt = 10