04-05 Water flowing into triangular trough

Problem 04
A triangular trough 10 ft long is 4 ft across the top, and 4 ft deep. If water flows in at the rate of 3 ft³/min, find how fast the surface is rising when the water is 6 in deep.

Solution 04

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Water flowing into trough with triangular sides and rectangular top

Volume of water:
$V = \frac{1}{2}xy (10) = 5xy$

By similar triangle:
$\dfrac{x}{y} = \dfrac{4}{4}$

$x = y$

$V = 5y^2$

$\dfrac{dV}{dt} = 10y \dfrac{dy}{dt}$

when y = 6 in or 0.5 ft
$3 = 10(0.5) \dfrac{dy}{dt}$

$\dfrac{dy}{dt} = 0.6 \, \text{ ft/min}$ answer

Problem 05
A triangular trough is 10 ft long, 6 ft wide across the top, and 3 ft deep. If water flows in at the rate of 12 ft³/min, find how fast the surface is rising when the water is 6 in deep.

Solution 05

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Volume of water:
$V = \frac{1}{2}xy (10) = 5xy$

By similar triangle:
$\dfrac{x}{y} = \dfrac{6}{3}$

$x = 2y$

$V = 5(2y)y = 10y^2$

$\dfrac{dV}{dt} = 20y \dfrac{dy}{dt}$

when y = 6 in or 0.5 ft
$12 = 20(0.5)\dfrac{dy}{dt}$

$\dfrac{dy}{dt} = 1.2 \, \text{ ft/min}$ answer

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