The diagram shows the dimensions of a polyhedron with parallel base and top - HSC - SSCE Mathematics Extension 2 - Question 9 - 2016 - Paper 1

To find the volume of the polyhedron, we need to consider the height $h$ at which the slice is taken parallel to the base.

The base dimensions of the polyhedron are as follows:

• The width at height 0 (base) is 8 units.
• The width at height 4 (top) is 2 units.

To express the volume, we will integrate the areas of the rectangles formed at different heights from 0 to 4:

1. Area of the slice: The area of the rectangle at height $h$ can be expressed as a function of $h$. Given the linear decrease in width from 8 at the bottom to 2 at the top, we can write the width at height $h$ as:

   $\text{Width at height } h = 8 - (h/4)(8-2) = 8 - \frac{3}{4}h$

   Therefore, the area becomes: $A(h) = (8 - \frac{3}{4}h) \cdot 4$

2. Setting up the integral: The volume $V$ can be obtained by integrating the area from height 0 to height 4:

   = \int_0^4 (8 - \frac{3}{4}h) \cdot 4 \, dh$$

3. Final expression: Thus, we arrive at the volume as the integral: $V = \int_0^4 \left(\frac{5}{4}h + 3\right) \cdot 2 \, dh$ This clearly matches option (A).