At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.

Verified step by step guidance

1. Start by expressing the relationship between the lengths of the legs of the triangle. Let the length of the short leg be x, then the long leg is 3x.
2. Use the formula for the area A of a right triangle, which is A = (1/2) * base * height. Here, base = x and height = 3x, so A = (1/2) * x * (3x).
3. Simplify the area formula to find A in terms of x: A = (3/2) * x^2.
4. Differentiate the area A with respect to time t using the chain rule: dA/dt = dA/dx * dx/dt, where dA/dx is the derivative of A with respect to x.
5. Calculate dA/dx from the area formula A = (3/2) * x^2, which gives dA/dx = 3x, and substitute this into the equation to relate dA/dt to dx/dt.