The area of a healing wound is given by $A=\pi r^{2}$. The radius is decreasing at the rate of 5 millimeter per day at the moment when $r=46$. How fast is the area decreasing at that moment?

Step 1 ：We are given that the area of a healing wound is given by the formula \(A=\pi r^{2}\), where \(r\) is the radius of the wound.

Step 2 ：We are also given that the radius is decreasing at the rate of 5 millimeters per day, which can be represented as \(\frac{dr}{dt}=-5\) mm/day.

Step 3 ：We are asked to find how fast the area is decreasing at the moment when the radius is 46 millimeters, or \(r=46\) mm.

Step 4 ：To find this, we can differentiate the equation for the area with respect to time, using the chain rule. The derivative of \(A\) with respect to \(t\) is \(\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}\).

Step 5 ：We can find \(\frac{dA}{dr}\) by differentiating \(A=\pi r^{2}\) with respect to \(r\), which gives us \(\frac{dA}{dr} = 2\pi r\).

Step 6 ：We can then substitute the given values for \(\frac{dr}{dt}\) and \(r\) into the equation to find \(\frac{dA}{dt}\).

Step 7 ：Substituting \(r=46\) and \(\frac{dr}{dt}=-5\) into the equation gives us \(\frac{dA}{dt} = 2\pi \cdot 46 \cdot -5 = -460\pi\).

Step 8 ：Final Answer: The area of the wound is decreasing at a rate of \(\boxed{-460\pi}\) square millimeters per day.