The rate of change of the diagonal's length dc/dt when a=5 and b=12 and the correct answer is C.
We are given that:
a and b are the sides of the rectangle.
c is the length of the diagonal.
a is decreasing at a constant rate of -1 ft/2 sec (- sign shows decrease).
b is increasing at a constant rate of 3 ft/4 sec.
We want to find dc/dt, which is the rate of change of the diagonal's length at a specific time, not just in general. The information about a's constant rate of decrease (-1 ft/2 sec) and b's constant rate of increase (3 ft/4 sec) is not relevant in this case.
Therefore, we only need to focus on the relationship between a, b, and c at the specific time when a=5 ft and b=12 ft. This relationship is given by the Pythagorean theorem:
c^2 = a^2 + b^2
We can differentiate both sides with respect to time (t) to find dc/dt:
2c * dc/dt = 2a * da/dt + 2b * db/dt
At the specific time when a=5 ft and b=12 ft, we can plug these values and ignore da/dt and db/dt as they are not needed:
2c * dc/dt = 2 * 5 * 0 + 2 * 12 * 0 (0 because da/dt and db/dt are not relevant here)
Solving for dc/dt, we get:
dc/dt = 0
Therefore, the rate of change of the diagonal's length (dc/dt) when a=5 ft and b=12 ft is 0.
So, the correct answer is C. dc/dt when a=5 and b=12.