A fish bowl is constructed out of a spherical bowl with diameter 20cm.
The bowl was initially empty and water flows into the bowl at a rate of 10cm^3/s. Let V and h be the volume and depth of water in the bowl at time t seconds respectively. Find dh/dt when h is increasing at the slowest rate.
The volume of a spherical cap of radius r and height h (the water depth) is
v = π/3 h^2(3r-h)
= π/3 h^2(30-h)
= π/3 (30h^2 - h^3)
dv/dt = π(20h - h^2) dh/dt
Now, when is h increasing most slowly? When the water surface area is greatest (since dv/dt is constant, and dv = a*dh)
That is when the bowl is half full: h=10.
10 = π/3 (20*10 - 100) dh/dt
dh/dt = 3/(10π) cm/s
Last few steps should be:
10 = π(20*10 - 100) dh/dt
dh/dt = 1/(10π) cm/s
thanks. leftover from a previous attempt.
To find dh/dt when h is increasing at the slowest rate, we need to find the point where the rate of change of h is at a minimum.
Given that the fish bowl is constructed out of a spherical bowl with a diameter of 20 cm, first, let's find the equation for the depth of water in terms of the volume of water.
The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3, where V is the volume and r is the radius.
Since the diameter is 20 cm, the radius is 10 cm.
V = (4/3) * π * (10 cm)^3
V = (4000/3) * π cm^3
Now, let's find the equation for the depth of water in the bowl, h, in terms of the volume of water, V.
The volume of water in the bowl can also be represented as the difference between the volume of the spherical bowl and the volume of the empty bowl.
Volume of water, V = Volume of the spherical bowl - Volume of the empty bowl
Volume of the empty bowl can be represented as the volume of the largest sphere that can fit within the bowl. Since the radius of the bowl is 10 cm, and the radius of the largest sphere that can fit within the bowl is half of that, i.e., 5 cm.
Volume of the empty bowl = (4/3) * π * (5 cm)^3
Volume of the empty bowl = (500/3) * π cm^3
Now, we can find the equation for the depth of water, h, in terms of the volume of water, V.
V = (4000/3) * π - (500/3) * π
V = (3500/3) * π cm^3
To find dh/dt, we need to differentiate the equation V = (3500/3) * π with respect to time t, using the chain rule.
dV/dt = (3500/3) * π * dh/dt
Since we know that water flows into the bowl at a rate of 10 cm^3/s (dm^3/s), we substitute dV/dt with 10.
10 = (3500/3) * π * dh/dt
Finally, we solve for dh/dt.
dh/dt = 10 / ((3500/3) * π)
Now we can calculate the value of dh/dt.