Question

What is the volume of water in a trough when the depth of the water is d?

A water trough is 6 m long and its cross-section is an isosceles trapezoid which is 100 cm wide at the bottom and 200 cm wide at the top, and the height is 50 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

`"see other solution"`

Explanation:

`"refer to Sean's solution"`

Answer 2

`V=600d^2+60000d` `cm^3`

Explanation:

.

The figure above shows the trough described in the problem. The front and back of the trough are isosceles trapezoids. The trough itself is a trapezoidal prism. The volume of prism is calculated by multiplying the area of the trapezoid `ABCD` by the length of the trough.

But we are asked to figure out the volume of the water in the trough, and the trough is not full.

The water level in the trough is shown by blue lines. The water in the trough forms a smaller trapezoidal prism whose length is the same as the length of the trough.

But the trapezoids in the front and the back of the water prism are smaller than those of the trough itself because the depth of the water `d` is smaller than the depth of the trough.

As the water level varies in the trough, `d` changes. This change affects the length of the large base of the trapezoids at both ends.

The volume of water is calculated by multiplying the area of trapezoid `CDHJ` by the length of the trough.

Since we have to find an expression for `V`, the volume of the water in the trough, that would be valid for any depth of water `d`, first we need to find an expression for the large base of trapezoid `CDHJ` in terms of `d` and use it to calculate the area of the trapezoid.

The large base is `HJ` which consists of three segments:

`HJ=HG+GK+KJ` `color(red)(Equation - 1)`

`GK`, in the middle, is equal to `DC` because `DE` and `CF` are drawn perpendicular to `GK` and `AB` which makes `CDGK` a rectangle. Therefore:

`GK=100 cm`

Since trapezoid `CDHJ` is also an isosceles trapezoid, the two right triangles `Delta CKJ` and `DeltaDHG` are congruent and

`HG=KJ`

For the same reason:

`AE=FB`

`AB=AE+EF+FB`

`EF=100cm`, `AB=200cm`

`200=AE+100+FB`

`200=AE+100+AE`

`200=2AE+100`

`2AE=100`

`AE=50cm`

THe right triangle `DeltaADE` is an isosceles right triangle because

`AE=DE=50cm`

Triangle `Delta DGH` is also an isosceles right triangle and

`DG=HG=d`

`KJ=HG=d`

Therefore, `color(red)(Equation - 1)` becomes:

`HJ=100+2d`

Area of a trapezoid `A=1/2h(b_1+b_2)`

Area of trapezoid `CDHJ` is:

`A=1/2d(100+100+2d)=1/2d(200+2d)=1/2d(2)(100+d)=d(100+d)`

`V=A(600)=600d(100+d)=60000d+600d^2` `cm^3`