Final answer:
The rate of change of the area of a right triangle with legs changing at different rates is found by using the formula for the area, resulting in an area that is decreasing at a rate of 5 square inches per second.
Step-by-step explanation:
A right triangle has legs of 3 inches and 4 inches that are changing over time, with the rate of change for the two legs being an increase of 5 in/sec for the shorter leg and a decrease of 10 in/sec for the longer leg. To find the rate of change of the area of this triangle, we use the formula for the area of a right triangle, which is A = 0.5 * base * height. Initially, the area is A = 0.5 * 3 * 4 = 6 in2. The rate of change of the area can be expressed as dA/dt = 0.5 * (dh/dt * base + db/dt * height), where dh/dt is the rate of change of the height (shorter leg) and db/dt is the rate of change of the base (longer leg).
Plug in the given rates of change to get dA/dt = 0.5 * (5 * 4 - 10 * 3) = 0.5 * (20 - 30) = 0.5 * (-10) = -5 in2/sec. Therefore, the area of the triangle is decreasing at a rate of 5 square inches per second.