The height of one solid limestone square pyramid is 12 m. A similar solid limestone square pyramid has a height of 15 m. The volume of the larger pyramid is 16,000 m3. Determine each of the following, showing all work and reasoning.(a) The scale factor of the smaller pyramid to the larger pyramid in simplest form(b) The ratio of the area of the base of the smaller pyramid to the larger pyramid(c) Ratio of the volume of the smaller pyramid to the larger(d) The volume of the smaller pyramid

Answer:Part a) The scale factor of the smaller pyramid to the larger pyramid in simplest form is [tex]\frac{4}{5}[/tex]Part b) The ratio of the area of the base of the smaller pyramid to the larger pyramid is [tex]\frac{16}{25}[/tex]Part c) The ratio of the volume of the smaller pyramid to the larger pyramid is [tex]\frac{64}{125}[/tex]Part d) The volume of the smaller pyramid is [tex]8,192\ m^{3}[/tex]Step-by-step explanation:Part a) The scale factor of the smaller pyramid to the larger pyramid in simplest formwe know thatIf two figures are similar, then the ratio of its corresponding sides is equal and this ratio is called the scale factorsoLetz----> the scale factorx----> the height of the smaller pyramidy----> the height of the larger pyramid   [tex]z=\frac{x}{y}[/tex]substitute the values[tex]z=\frac{12}{15}[/tex]Simplify[tex]\frac{12}{15}=\frac{4}{5}[/tex] -----> scale factor in simplest formPart b) The ratio of the area of the base of the smaller pyramid to the larger pyramidwe know thatIf two figures are similar, then the ratio of its areas is equal to the scale factor squaredsoLetz----> the scale factorx----> the area of the base of the smaller pyramidy----> the area of the base of the larger pyramid   [tex]z^{2} =\frac{x}{y}[/tex]we have[tex]z=\frac{4}{5}[/tex]substitute[tex](\frac{4}{5})^{2} =\frac{x}{y}[/tex][tex](\frac{16}{25})=\frac{x}{y}[/tex]Rewrite[tex]\frac{x}{y}=\frac{16}{25}[/tex] -----> ratio of the area of the base of the smaller pyramid to the larger pyramidPart c) Ratio of the volume of the smaller pyramid to the larger pyramidwe know thatIf two figures are similar, then the ratio of its volumes is equal to the scale factor elevated to the cubesoLetz----> the scale factorx----> the volume of the smaller pyramidy----> the volume of the larger pyramid   [tex]z^{3} =\frac{x}{y}[/tex]we have[tex]z=\frac{4}{5}[/tex]substitute[tex](\frac{4}{5})^{3} =\frac{x}{y}[/tex][tex](\frac{64}{125})=\frac{x}{y}[/tex]Rewrite[tex]\frac{x}{y}=\frac{64}{125}[/tex] -----> ratio of the volume of the smaller pyramid to the larger pyramidPart d) The volume of the smaller pyramidwe know thatIf two figures are similar, then the ratio of its volumes is equal to the scale factor elevated to the cubesoLetz----> the scale factorx----> the volume of the smaller pyramidy----> the volume of the larger pyramid   [tex]z^{3} =\frac{x}{y}[/tex]we have[tex]z=\frac{4}{5}[/tex][tex]y=16,000\ m^{3}[/tex]substitute and solve for x[tex](\frac{4}{5})^{3} =\frac{x}{16,000}[/tex][tex](\frac{64}{125})=\frac{x}{16,000}[/tex][tex]x=16,000*64/125[/tex][tex]x=8,192\ m^{3}[/tex]