To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let's denote the horizontal leg as x and the vertical leg as y. We are given that dx/dt (the rate at which x is changing) is 5 in./min and dy/dt (the rate at which y is changing) is -6 in./min. We are asked to find dh/dt (the rate at which the hypotenuse h is changing) when x = 12 in. and y = 9 in.

First, let's use the Pythagorean theorem to express h in terms of x and y:

h^2 = x^2 + y^2

Differentiating both sides of this equation with respect to time t, we get:

2h * dh/dt = 2x * dx/dt + 2y * dy/dt

Simplifying this equation, we have:

h * dh/dt = x * dx/dt + y * dy/dt

Substituting the given values, we have:

h * dh/dt = 12 in. * 5 in./min + 9 in. * (-6 in./min)

h * dh/dt = 60 in/min - 54 in/min

h * dh/dt = 6 in/min

Now, we can solve for dh/dt:

dh/dt = 6 in/min / h

Substituting the given values for x and y into the Pythagorean theorem, we have:

h^2 = 12^2 + 9^2

h^2 = 144 + 81

h^2 = 225

h = 15

Substituting this value into the equation for dh/dt, we have:

dh/dt = 6 in/min / 15 in.

dh/dt = 2/5 in/min

Therefore, the rate at which the hypotenuse is changing when the horizontal leg is 12 in. and the vertical leg is 9 in. is 2/5 in/min. Therefore, the correct answer is (a) 2/5 in/min.

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