To find the value of db/dt, let's use the formula for the area of a right triangle:
Area = (1/2) * base * height
In this case, the base (b) is increasing with respect to time (t), and the rate of change of b with respect to time is db/dt.
Given that dc/dt = 1, we know that the rate of change of the hypotenuse is 1. Since c is the hypotenuse, and in a right triangle, c^2 = a^2 + b^2, we can differentiate both sides with respect to time:
2c * (dc/dt) = 2a * (da/dt) + 2b * (db/dt)
Substitute dc/dt = 1:
2c = 2a * (da/dt) + 2b * (db/dt)
Now, let's substitute the given values: b = 4 and Area = 6.
Area = (1/2) * b * a
6 = (1/2) * 4 * a
a = 3
Now, plug in a = 3 and b = 4 into the equation:
2c = 2 * 3 * (da/dt) + 2 * 4 * (db/dt)
Since dc/dt = 1, we have:
2 = 6 * (da/dt) + 8 * (db/dt)
Now, solve for db/dt:
8 * (db/dt) = 2 - 6 * (da/dt)
Given that da/dt = 3 (as given in the question), substitute this value:
8 * (db/dt) = 2 - 6 * 3
8 * (db/dt) = 2 - 18
8 * (db/dt) = -16
Now, solve for db/dt:
(db/dt) = -2
Therefore, the value of db/dt is -2.
