Imade a fort for my pet gorilla by connecting two boxes. the first box is 555 meters long, 999 meters wide, and 999 meters high. the second box is 333 meters long, 888 meters wide, and 222 meters high. how many cubic meters of space does my gorilla have to play in her fort?
we know that
The total cubic meters of space is equal to the volume of the first box plus the volume of the second box
The volume of the box is equal to
Find the volume of the box 1
Find the volume of the box 2
The total volume is equal to
we turn each of our sides into direction vectors
ab=b-a=(3 - 7, 1 - 0) = (-4, 1)
we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram. we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2. so it's not a square or a rhombus.
we check the dot product of direction vectors of adjacent sides. we have
ab dot bc = -4(1) + 1(6) = 2
that's not zero so the sides are not perpendicular. we've ruled out rectangle.
the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.
we need to solve for 'x' the following equation:
we have that:
factorizing, we get that:
then, the solutions are: or
nothing extraneous has been found!