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Question

A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet.

(a) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when h is 1.8 feet deep?

___________ ft/min

(b) If the water is rising at a rate of 3/8 inch per minute when h = 2.2, determine the rate at which water is being pumped into the trough.

__________ft³/min

Answer 1

Let

l = 12 ft be the length of the trough,

v be the rate of pumping (in part (a) given, but in part (b) to be calculated),

t be time variable,

h be the water height variable.

First we calculate the volume V of water for any water height h.

Notice that the isosceles triangle forming the base of the through

is similar to the isosceles triangle forming the base of the volume of water.

Since the base and height of the former are equal, so is

the base and height of the latter.

Therefore the water forms a trough shape of length l,

ending in a triangle with base and height both h.

Therefore

V = l h²/2 (1)

Over time t the water can fill volume:

V = v t. (2)

Combining (1) and (2)

v t = l h² / 2

From the above equation,

we will need h expressed as function of t and also t expressed as function of h:

t = l h² / (2 v) (3)

h = sqrt(2 v t / l) (4)

From (4), the derivative h'(t) = dh/dt is

h'(t) = v / (l sqrt(2 v t / l))

(a)

We are to calculate the derivative h'(t₁) at a time t₁ when

h₁ = 1.8 ft

From (3) that happens at time

t₁ = l h₁² / (2 v)

To evaluate the derivative at time t₁ it is convenient to first

substitute t₁ into the expression (2 v t₁ / l) under the square root:

2 v t₁ / l = 2 v (l h₁² / (2 v)) / l = h₁²

Now

h'(t₁) = v / (l sqrt(2 v t₁ / l) = v / (l h₁) (5)

Substituting actual numbers:

h'(t₁) = 2 ft³/min /(12 ft x 1.8 ft) = 0.09 ft/min

(b)

We are given h'(t₂) = 3/8 in/min at a time t₂ when the water height is h₂ = 2.2 ft,

and we are to calculate v.

Reusing equation (5):

h'(t₂) = v / (l h₂)

v = l h₂ h'(t₂)

Substituting actual numbers

v = 12 ft x 2.2 ft x 3/8 in/min

= 12 x 12 in x 2.2 x 12 in x 3/8 in/min = 1425.6 in³/min