A water trough is 4 m long and its cross-section is an isosceles trapezoid which is 210 cm wide at the bottom and 280 cm wide at the top, and the height is 70 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.

Answer 1

make a sketch of the cross-section
draw verticals from the ends of the base , to get two identical right angled triangles and a rectangle
the height of the triangle will be 70 and the top will be 35. The rectangle will be 210 by 70

Draw a water-line , let the part inside the triangle be x
and let the height of the water be d

Let's concentrate on one of the triangles.
by ratios:
x/d = 35/70
x/d = 1/2
x = d/2

Volume of water = 400( 2 triangles + rectangle)
= 400( 2(1/2)xd + 210(70) )
= 400 xd + 5880000
= 400(d/2)(d) + 5880000
= 200d^2 + 5880000 , in cm^3