6. A parabaloid with height h and radius r has volume 1/2 �pieR^2H. Consider a parabaloid-shaped water
tank with height of 7m and radius of 4m. If V is the volme of water in the tank and h is the depth
of water,
find the relationship between dV/dt and dhdt.
How is the volume changing when h = 3m and
dh/dt = 0.2m/s?
dV/dt = surface area * dh/dt (geometry , no calculus really required, draw picture)
dV/dt = pi r^2 dh/dt
consider a cross-section of the paraboloid placed with vertex at the origin an opening upwards
then its equation would be
y = ax^2
when x=4, y = 7
y = (7/16)x^2
or
h = (7/16)r^2 , using r and h
r^2 = 16h/7
V = (1/2)πr^2h
= (1/2)π(16h/7)h
= (8π/7) h^2
dV/dt = 16π/7 h dh/dt ----> the relationship between dV/dt and dh/dt
Whe h=3 and dh/dt = .2
dV/dt = (16π/7)(3)(.2) = ......
I will let you do the button pushing.
To find the relationship between dV/dt and dh/dt, we need to differentiate the equation for the volume of the paraboloid with respect to time.
Given that the volume of the paraboloid is V = (1/2)πr^2h, where r is the radius and h is the height, we can differentiate it with respect to time to find dV/dt.
To do that, we need to use the chain rule. Let's assume that both r and h are functions of time. Then the volume V is also a function of time, and we can write it as V(t).
Differentiating both sides of the equation V = (1/2)πr^2h with respect to time, we get:
dV/dt = (1/2)π [(2r d(r)/dt)h + r^2(dh/dt)]
Now we have the relationship between dV/dt and dh/dt. It involves the rates at which both the radius and the height are changing with respect to time.
Next, let's find how the volume is changing when h = 3m and dh/dt = 0.2m/s.
Let's use the equation we derived earlier: dV/dt = (1/2)π [(2r d(r)/dt)h + r^2(dh/dt)]
Substitute the given values: h = 3m and dh/dt = 0.2m/s. We need to find the value of dV/dt.
At h = 3m, we need to know the value of r, the radius. Without that information, we cannot determine the exact value of dV/dt.
Please provide the value of the radius r at h = 3m, and I can help you find the value of dV/dt.