# Question 0b699

Oct 19, 2015

Use the formula for a prismatoid .

#### Explanation:

The shape of the ore pile is called a prismatoid.

The long side has a base of 500 ft, but the top narrows down to 440 ft. So the shape of this face (and the opposite side) of the prismatoid is a trapezoid .

Now, the narrow side has a base of 60 ft and forms a triangle with a height of 30 feet.

The four faces (two triangles and two trapezoids) meet at the top of the prismatoid to form a straight line that runs parallel to the 500 ft edges. The formula for the volume of a prismatoid is:

$V = \frac{1}{6} \left[B + \left(4 M\right) + b\right] \cdot h$

where,

$B =$ the area of the base $= 60 \cdot 500 = 30 , 000 f {t}^{2}$
$b =$ the area of the top $= 0$ (since it is a straight line)
$M =$ the area of the middle section $= 470 \cdot 30 = 14 , 100 f {t}^{2}$
$h =$ the height =30 ft^

[ Note : the middle area (M) forms a cross-section in the shape of a rectangle. The ends with the triangles have a base of 60, so halfway up the triangle the line parallel to the base has length 30 ft . Similarly, the trapezoid has a line parallel to the base and halfway up with length $= \frac{1}{2} \left(440 + 500\right) = 470 f t$.]

Plugging these all into to volume formula:

$V = \frac{1}{6} \left[30 , 000 + \left(4 \cdot 14 , 100\right) + 0\right] \cdot 30 = 432 , 000 f {t}^{2}$

=432,000ft^2*(110 lbs)/(ft^2)/((2000 lbs) = 23,760# tons