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.
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
To find the volume of water in the trough, we need to calculate the area of the cross-section and then multiply it by the length of the trough.
The cross-section of the trapezoid can be divided into two parts: the smaller rectangle at the bottom and the trapezoidal portion at the top.
To calculate the area of the trapezoidal portion, we can use the formula:
Area = (1/2) * (a + b) * h
where:
a = width of the bottom (210 cm)
b = width of the top (280 cm)
h = height of the trapezoid (70 cm)
Substituting the given values into the formula, we get:
Area = (1/2) * (210 cm + 280 cm) * 70 cm
= (1/2) * 490 cm * 70 cm
= 17150 cm^2
Now, since the depth of the water is d cm, the area of the rectangular portion can be calculated as:
Area of the rectangle = width * depth
= 210 cm * d cm
= 210d cm^2
To find the total volume of water in the trough, we multiply the sum of these two areas by the length of the trough:
V = Length * (Area of trapezoid + Area of rectangle)
= 4 m * (17150 cm^2 + 210d cm^2)
However, the units need to be consistent, so we need to convert the length from meters to centimeters:
1 m = 100 cm
V = 400 cm * (17150 cm^2 + 210d cm^2)
= 684,000 cm^3 + 84,000d cm^3
Therefore, the expression for the volume of water in the trough, V, when the depth of the water is d cm is:
V = 684,000 cm^3 + 84,000d cm^3