Question 135897
{{{A =B+ps/2}}} is the correct formula for the surface area of a pyramid where {{{B}}} is the area of the base, p is the perimeter of the base, and s is the slant height along a bisector of a face (measured from the midpoint of any edge to the apex of the pyramid).  However, you have a pyramid that is part of a complex solid with the base of the pyramid mated to one base of the square based prism.  Therefore the area of the base should not be included in the calculation of the surface area of the pyramid part of the solid.  Hence, {{{A=ps/2}}}.  Since the pyramid has a base that matches the square base of the square based prism, each edge of the pyramid base measures 6m.  So the perimeter of the base is {{{4*6=24m}}}.  Putting it together {{{(24*5)/2=60m^2}}}


{{{A=2B+ph}}} is the formula for the surface area of a prism where B is the area of the base, p is the perimeter of the base and h is the height of the prism.  But again, you have to eliminate the area of one of the bases because one of them mates to the bottom of the pyramid.  So the applicable formula is: {{{A=B+4bh}}}.  The area of the base is simply the area of a 6 by 6 square, or {{{36m^2}}}, and the perimeter of the base is the same as that for the pyramid, namely 24m.  So:


{{{36+24(8)=228m^2}}}


Adding the two surface areas together, {{{228+60=288m^2}}}


The volume of a pyramid is given  by {{{V=(1/3)Bh}}} where B is the area of the base and h is the height of the pyramid which is the measure of a segment perpendicular to the base through the apex.  We are given the slant height, but the slant height and one-half of the base edge measure forms a right triangle with the height, the slant height being the hypotenuse.  Therefore, the height of the pyramid is {{{sqrt(5^2-3^2)=sqrt(16)=4}}}.  The area of the pyramid base is the same as the area of the prism base, namely {{{36m^2}}}, so the volume of the pyramid is {{{V=(1/3)36*4=12*4=48m^3}}}


The volume of a prism is given by {{{V=Bh}}} where B is the area of the base and h is the height.  We already know the area of the base and we are given the height, so:  {{{V=36*8=288m^3}}}


Together {{{V[T]=V[Pr]+V[Py]=288+48=336m^3}}}