Math CentralQuandaries & Queries


Question from Jasmine, a student:

A water trough is 8 m long and its cross-section is an isosceles trapezoid which is 90 cm wide at the bottom and 120 cm wide at the top, and the height is 30 cm. The trough is not full. Give an expression for V , the volume of water in the trough in cm^3, when the depth of the water is d cm.

Hi Jasmine.

The trapezoid representing the cross section of the water has therefore a base of 90 cm wide (b), a depth of d cm and a top of how long (t)? That's the only tricky step because then you can use the formula for the area of the (water's) trapezoid (t + b)d/2 and just multiply by the length 800cm to get the volume in cm3.

Because the sides of the troughs are straight lines, the top width of the water in the trough is linearly related to the depth and the width variation from the top to the bottom of the trough. In other words, the top width of the water at depth = 30cm is 120cm and the top width of the water at depth = 0cm is 90cm. If d = depth and t = top width of the water, then the top width of the water is related to d by the relationship (d, t) where (0, 90) and (30, 120) are two points on that line (linear relation). If you write out this as an equation in the form t = f(d), then you have your expression for the top width of water.

Hope this helps. Write back if you need an example or more help,
Stephen La Rocque

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