You have a conical tank, vertex down, which is 12 feet across the top and 18 feet deep. If water flows in at a rate of 9 cubic feet per minute, find the exact rate of change when the water is 6 feet deep.
You know the rate of dV/dt (inflow), and you can get the volume of a cone (1/3 h * toparea). So the trick is to write an equation relating top area to h (ie: toparea= PI*(12h/18)^2 /144 ) check that.
take the derivative of V with respect to h, and solve.
To find the exact rate of change, we need to relate the variables involved and then differentiate the equation.
Let's start by deriving the equation that relates the top area of the conical tank to its height, h. We know that the top area of the cone is given by the formula A = π(r^2), where r is the radius at a particular height h.
Given that the diameter of the top of the conical tank is 12 feet, the radius, in terms of h, would be (12h/18), as the height and diameter are proportional. Therefore, the equation for the top area of the tank in terms of h is:
A = π * ((12h/18)^2)
We also know that the volume of a cone is given by the formula V = (1/3) * A * h. Substituting the equation for A, we can rewrite the volume equation in terms of h:
V = (1/3) * π * ((12h/18)^2) * h
Now, we need to find the exact rate of change (dV/dt) when the water is 6 feet deep, which means finding dV/dh and then multiplying by the rate of change of h with respect to time.
To find dV/dh, we differentiate the volume equation with respect to h:
dV/dh = (1/3) * π * (2 * (12h/18) * (12/18) * h + ((12h/18)^2))
Simplifying this equation, we get:
dV/dh = (1/3) * π * (16h^2/9 + 4h^2)
dV/dh = (4/3) * π * ((16h^2 + 12h^2) / 9)
dV/dh = (4/3) * π * (28h^2 / 9)
Now, to find the exact rate of change, we need to evaluate dV/dh at h = 6 (since the water is 6 feet deep), which gives:
dV/dh = (4/3) * π * (28*(6^2) / 9)
Finally, you can compute the exact rate of change by multiplying the above value by the rate at which the water is flowing into the tank, which is 9 cubic feet per minute:
Exact rate of change = (4/3) * π * (28*(6^2) / 9) * 9
Note: Make sure to evaluate the expression using the appropriate units to get the final answer.