Final answer:
To find the rate of change of the height of the cone, we can differentiate the equation for the volume of a cone with respect to time and solve for the rate of change of the height. Substituting the given values, we can find the rate of change of the height.
Step-by-step explanation:
To find the rate of change of the height of the cone, we can use the formula for the volume of a cone: V = πr²h, where V is the volume, r is the radius, and h is the height. Now, we know that the radius is increasing at a constant rate of 3 inches per minute, so we can write r = 3t + 1, where t is the time in minutes. We also know that the volume is increasing at a rate of 43 cubic inches per minute, so we can write V = 43t + 13, where t is the time in minutes. Finally, we can substitute these expressions into the formula for the volume of a cone and solve for the rate of change of the height.
V = π(3t + 1)²h = 43t + 13
π(3t + 1)²h = 43t + 13
(9πt² + 6πt + π)h = 43t + 13
9πt²h + 6πth + πh = 43t + 13
9πt²h + 6πth + (πh - 43t - 13) = 0
Now, we can differentiate this equation with respect to time t and solve for the rate of change of the height dh/dt.
18πth + 6πh + 43 = 0
18πth + 6πh = -43
We are given that the radius of the cone is 1 inch and the volume is 13 cubic inches at the instant we are interested in. So we can substitute these values into the equation and solve for the rate of change of the height dh/dt.
18πt + 6πh = -43
18πt + 6π = -43
18πt + 6π = -43
Now we can solve this equation to find the value of t.
18πt = -49
t = -49/18π
Now we can substitute this value of t back into the equation for the rate of change of the height to get our answer.
dh/dt = 18π(-49/18π) + 6πh = -49 + 6πh