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For a plot of land of 100 m × 80 m, the level is to be raised by spreading the earth from a stack of a rectangular base 10 m × 8 m with vertical section being a trapezium of height 2 m. The top of the stack is 8 m × 5 m. By how many centimetres can the level be raised?

I know the approach needed to solve this problem. I only confusion I have about this problem is the calculation of the volume of the stack which I believe is the trapezoidal prism (or truncated (right) rectangular prism or frustum of (right) rectangular prism). I also assume a prism is the same thing as a pyramid for geometrical purposes.

A trapezoidal prism is a 3D figure made up of two trapezoids that is joined by four rectangles.

I saw online different methods giving different answers to this question. I am confused what is the correct approach.

Method I
enter image description here
Volume of Right rectangular pyramid

enter image description here

Method II

enter image description here Identify the parallel sides of the base (trapezoid) to be $b_{1}$ and $b_{2}$ and the perpendicular distance between them to be $h$ and find the area of the trapezoid using the formula:
Area of the trapezoid $=\frac{1}{2}\left(b_{1}+b_{2}\right) \times h$
Identify its height / length of the prism (the vertical distance between two bases).
Multiply the base area and the height to find the volume. enter image description here

Method III
enter image description here

$$ \text { Volume of truncated pyramid/prism }=\frac{1}{3}\left(A_{1}+A_{2}+\sqrt{A_{1} A_{2}}\right) h $$ $$ A_{1} = 8X5 = 40 \mathrm{m}^{2} $$ $$ A_{2} = 10X8 = 80 \mathrm{m}^{2} $$ $$ \Rightarrow\text { Volume of truncated pyramid/prism }=\frac{1}{3}\left(40+80+\sqrt{40X80}\right) 2 $$ $$ \Rightarrow\text { Volume of truncated pyramid/prism }= 58.85 \mathrm{m}^{3} $$

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    $\begingroup$ Method II is the right one: a prism is NOT a pyramid. $\endgroup$ Dec 28, 2021 at 17:45
  • $\begingroup$ @Intelligentipauca In the original problem statement, it is not specified whether to consider prism or pyramid. How to know which shape to select? $\endgroup$
    – Anubhav
    Dec 28, 2021 at 17:49
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    $\begingroup$ They say: "with vertical section being a trapezium of height 2 m". If all sections are equal, it's a prism. And even if they weren't equal, it's enough they have the same area. $\endgroup$ Dec 28, 2021 at 17:51
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    $\begingroup$ @Andrei If it were a pyramidal frustum then it couldn't have all its vertical sections formed by a trapezium of height 2 m. $\endgroup$ Dec 28, 2021 at 17:53
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    $\begingroup$ @Intelligentipauca you are right. It looks more like the mansard roof in this image $\endgroup$
    – Andrei
    Dec 28, 2021 at 18:06

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The pyramid-based answers do not work because the trapezoidal prism is not actually part of a pyramid: the non-horizontal edges do not meet in a single point. Furthermore the question might be ambiguous whether the $8m$ edge of the top face is parallel or perpendicular to the $8m$ edge of the bottom face, and this affects the final result.

In case the $8m$ on top and bottom are parallel, you have a trapezium prism, with trapezium area $(10m+5m)/2 \times 2m$ and "height" $8m$ (perpendicular to the trapezium), resulting in a volume of $120 m^3$.

If the top and bottom faces of the stack are laid out as hinted in the question, with the bottom $10m$ parallel to the top $8m$ and the bottom $8m$ parallel to the top $5m$, it is neither a trapezium prism nor a truncated pyramid, because the non-horizontal edges do not intersect in a single point. Assuming the faces are still plane, the cross-section at height $x$ (measured in $m$) is given by $(10-x)\times(8-\frac 32 x)$, and the volume can be determined by integration to yield $V = \int_0^2(10-x)(8-\frac 32 x)dx = 118 m^3$.

In general the formula to compute such a shape with height $h$, top rectangle $a\times b$ and bottom rectangle $c \times d$, with the $a$ side parallel to the $c$ side, is $\frac16 h(2ab+2cd+ad+bc)$. I'm not sure whether you are expected to know this though.

Here is a Blender model of the second case, demonstrating how it is not a frustum:

enter image description here

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    $\begingroup$ The second case must be discarded, because all vertical sections of the stack are formed by a trapezium of height 2 m. And, by the way, your second case IS a pyramidal frustum. $\endgroup$ Dec 28, 2021 at 18:00
  • $\begingroup$ The question says a vertical section of the stack is such a trapezium, not all of them (in which case the posed scenario would be impossible in both cases). $\endgroup$
    – Magma
    Dec 28, 2021 at 18:03
  • $\begingroup$ @Intelligentipauca I have added an image to demonstrate that the second case is indeed not a pyramidal frustum. $\endgroup$
    – Magma
    Dec 28, 2021 at 18:04
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    $\begingroup$ You are right about the frustum, but the text reads "a stack with vertical section being a trapezium of height 2 m" and not "a stack with a vertical section being a trapezium of height 2 m". $\endgroup$ Dec 28, 2021 at 18:07
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    $\begingroup$ @Intelligentipauca I did not do any modifications from my end. That is the original text. $\endgroup$
    – Anubhav
    Dec 28, 2021 at 18:14

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