Looking for help on this

A spherical water storage container with a diameter of 1051 m is filled with water at
a rate of 30 m^3/s. At what rate is the area of the surface of the water changing when the
height of the water is 800 m from the bottom of the tank?

Question

Looking for help on this

A spherical water storage container with a diameter of 1051 m is filled with water at
a rate of 30 m^3/s. At what rate is the area of the surface of the water changing when the
height of the water is 800 m from the bottom of the tank?

whpalmer4 · Answer

Can you form an expression for the surface area of the water as a function of the height of the water?

anonymous · Answer

I hate this section and honestly don't know where to begin. So no.

anonymous · Answer

sec, I'll take a stab

whpalmer4 · Answer

well, what's the formula for the (surface) area of a circle?

anonymous · Answer

4pir^2

whpalmer4 · Answer

Right.  Now the radius is going to be a function of the height of the water, right?

At this point, a good picture is probably going to be useful.

whpalmer4 · Answer

Actually, no 4 in there, that would be the surface area of a sphere, we just want the area of the circle which forms the surface of the water.  That's going to be $A = \pi r^2$, right?

anonymous · Answer

right

whpalmer4 · Answer

|dw:1396985733288:dw|  Cross-sectional view of the tank

anonymous · Answer

$$\pi r^{2}$$$$\frac{ dh }{ dt }$$

anonymous · Answer

$$\pi(525.5)^{2}(30)$$

whpalmer4 · Answer

my diagram is a bit misleadingly labeled — I don't think the radius of the surface area is 1051 when the height of the water is 800

whpalmer4 · Answer

that 1051 is referring to the radius of the tank, with the { drawn in

whpalmer4 · Answer

If we look at the sphere with the bottom point as (0,0), 

|dw:1396986991965:dw|

whpalmer4 · Answer

ack, that's not right.  formula should be $x^2 + (y-1051)^2 = 1051^2$ because the center of that circle is at (0,1051)