![]() To do this derivation, we have to use the chain rule on the right hand side: Take the derivative of the equation with respect to time. And because the volume of water ( V) is equal to this cross-sectional area times the length of the trough, then we have an equation relating the volume of water to the depth ( h) of water:Ä¢. Since the area of the isosceles triangle is xh, this equals ( h/4) h = h 2/4. So if we know h, we know x (and vice versa). The ratios of corresponding sides of similar triangles are equal. We can use the principle of similar triangles to relate x to h though: ![]() The area of the isoceles triangle filled with water is xh. The cross section is an isosceles triangle, of course, whose shape is defined by the relative sizes of its sides (these are given). The volume of the water in the trough equals the length of the trough times the cross-sectional area of the trough up to the depth it is filled with water.
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