so they want you to solve for each trapezoid (base1 + base 2)/2 × height. and solve for the one triangle, 1/2 × base × height. add them all together..... or add all the bottom lengths (the 5's) together and multiply by you height (12) times 1/2 (the triangle formula).
Because logarithms were a lot less time-consuming than other methods of finding roots. They were also more accurate in the sense that it was harder to make a mistake.
