A conical tank has height 3 m and radius 2 m at the top. Water flows in at a rate of 1.4 \text{m}^3\text{/min}. How fast is the water level rising when it is 2.6 m?
Oh, it is pointed at the bottom?
area = pi r^2
radius = (1/3) h for 1 at the top and 0 at the bottom
so surface area A = pi (1/9)h^2
for h = 2.6, A = 2.36 meters^2
d V = A dh
dV/dt = A dh/dt or dh/dt = (1/A)dV/dt
dV/dt given as 1.4 m^3/min
so
dh/dt = 1.4/2.36 = .593 m/min
To find the rate at which the water level is rising, we need to use related rates calculus.
Let's denote the height of the water in the conical tank as "h" (in meters) and the radius of the water at height "h" as "r" (in meters). We are given that the height of the tank is 3 m and the radius at the top is 2 m.
First, we need to find an equation that relates the height "h" and the radius "r" of the water. Since the tank is conical, we can use the similar triangles property:
(r / R) = (h / H)
Where "R" is the radius of the top of the tank and "H" is the height of the tank. Substituting the given values, we have:
(2 / 2) = (h / 3)
Simplifying, we get:
1 = h / 3
h = 3
Now, we have an equation that relates the height "h" of the water and the radius "r":
r = (h / 3) * R
Next, we differentiate both sides of the equation with respect to time to relate the rates of change:
dr/dt = (1 / 3) * R * dh/dt
Here, dr/dt represents the rate at which the radius "r" is changing with respect to time, and dh/dt represents the rate at which the height "h" is changing with respect to time.
We are given that water flows into the tank at a rate of 1.4 m^3/min. To relate this to the rate of change of the height, we can use the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
Differentiating both sides with respect to time, we get:
dV/dt = (2/3) * π * r * (dr/dt) * h + (1/3) * π * r^2 * (dh/dt)
We can substitute the given values and known values of "r" and "dh/dt" into this equation:
1.4 = (2/3) * π * 2 * (dr/dt) * 2.6 + (1/3) * π * 2^2 * dh/dt
Now we can solve for dh/dt, the rate at which the water level is rising when it is 2.6 m.