Cylinder Tank

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Draining a half-cylinder tank calculus problem please help!!!!!?

A trough is shaped like a half-cylinder with length 5 m and radius 1 m. The tanks full of water when a valve is opened and water flows out of the bottom of the tank at a rate of 1.5m^3/hr. (the area of a sector of a circle of radius r subtended by an angle theta is r^2 theta/2.)

A. How fast is the water level changing when the water level is 0.5m From the bottom of the trough?

B.what is the rate of change of the surface area of the water when the water is 0.5m deep?

A.
The cross-section of the liquid in the tank is a spherical segment of height h (= the depth of the liquid)
(for a sketch see [1]

The total volume of liquid equals cross-sectional area time the length of the cylinder:
V = A∙L
The volume of decreases by a constant rate Q = 1.5 m³∙hr⁻¹, i.e.
dV/dt = -Q
When you expand left hand side using chain rule, you get:
d(A∙L)/dt = L∙(dA/dt) = L∙(dA/dh)∙(dh/dt) = -Q
So the rate of change of the liquid level is given by:
dh/dt = -Q / (L∙(dA/dh)
So when you know dA/dh as function of h you can compute the change of the liquid level at depth h.

The area of the segment is:
A = R²∙cos⁻¹((R-h)/R) - (R - h)∙√(2∙R∙h - h²)
= R²∙cos⁻¹((1 -(h/R)) - (R - h)∙√(2∙R∙h - h²)
(R is the radius of the circle)
=>
(dA/dh) = [R²∙(-1/R)∙(1/√(1 - ((1 -(h/R))²)] + √(2∙R∙h - h²) - [(R - h)∙(1/2)∙(2∙R - 2∙h)/√(2∙R∙h - h²)]
= [R/√(1 - 1 + 2∙(h/R) - (h/R)²)] + √(2∙R∙h - h²) - [(R - h)²/√(2∙R∙h - h²)]
= [R/√(2∙(h/R) - (h/R)²)] + √(2∙R∙h - h²) - [(R² - 2∙R∙h + h²) /√(2∙R∙h - h²)]
= (R² + (2∙R∙h - h²) - (R² - 2∙R∙h + h²)) / √(2∙R∙h - h²)
= 2∙(2∙R∙h - h²) / √(2∙R∙h - h²)
= 2∙√(2∙R∙h - h²)

Hence,
dh/dt = -Q / (2∙L∙√(2∙R∙h - h²))
= - 1.5 m³∙hr⁻¹ / (2 ∙ 5m ∙ √(2∙ 1m ∙ 0.5m - (0.5m)²))
= -0.173 m∙hr⁻¹

B.
The area of the liquid surface equals the length of the chord, which bounds the circular segment times the length of the cylinder:
S = a∙L
=>
dS/dt = L∙(da/dt) = L∙(da/dh)∙(dh/dt)

with
a = 2∙√(2∙R∙h - h²)
=>
da/dh = (2∙R - 2∙h)/√(2∙R∙h - h²) = 2∙(R - h)/√(2∙R∙h - h²)
=>
dS/dt = L ∙ [2∙(R - h)/√(2∙R∙h - h²)] ∙ [- Q/(2∙L∙√(2∙R∙h - h²))]
= - Q ∙ (R - h)/(2∙R∙h - h²)
= - 1.5 m³∙hr⁻¹ ∙ (1m - 0.5m)/(2∙1m∙0.5m - (0.5m)²)
= - 1 m²∙hr⁻¹

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