Answer:
[tex]\text{a. }128\:\mathrm{cm^3},\\\text{b. }4:1,\\\text{c. }179.4\:\mathrm{cm^2};\:44.8\:\mathrm{cm^2}[/tex]
Step-by-step explanation:
The volume of a square pyramid is given by [tex]V=\frac{1}{3}\cdot s^2\cdot h[/tex]
Since pyramids A and B are similar, their corresponding side lengths are proportion. Since the base edge of B is half that of A's, each dimension of B will be half of A. Since the formula for volume requires the multiplication of three dimensions, the volume of pyramid A will be [tex]2^3=8[/tex] times larger than the volume of pyramid B.
Thus, the volume of pyramid A is equal to [tex]16\cdot 8=\boxed{128\:\mathrm{in^3}}[/tex]. You can also find the dimensions of pyramid A and use the formula.
The surface area consists of the sum of all areas of the 2D shapes that make the figure. Since all dimensions of pyramid A are twice the dimensions of pyramid B, the ratio of the surface area of pyramid A to pyramid B is [tex]2^2:1=\boxed{4:1}[/tex]
The surface area of a square pyramid can be found by adding the areas of the three triangles and one square that make it up. As one long messy formula, that becomes [tex]2s\sqrt{\left(\frac{s}{2}\right)^2+h^2}+s^2[/tex] for a square pyramid with base edge [tex]s[/tex] and height [tex]h[/tex].
Plugging in values, we get the following:
[tex]A_a\approx \boxed{179.4\:\mathrm{cm^2}},\\A_b\approx \boxed{44.8\:\mathrm{cm^2}}[/tex]
