To determine how the volume of a square pyramid changes when the base area is quadrupled and the height is reduced to one-third of its original size, let us walk through the problem step-by-step.
1. Understand the Volume Formula for a Pyramid:
The volume [tex]\( V \)[/tex] of a square pyramid is given by:
[tex]\[
V = \frac{1}{3} B h
\][/tex]
where [tex]\( B \)[/tex] is the area of the base and [tex]\( h \)[/tex] is the height.
2. Define Original Dimensions:
Let the original base area be [tex]\( B \)[/tex] and the original height be [tex]\( h \)[/tex].
3. New Dimensions with Given Conditions:
- The base area is quadrupled: new base area [tex]\( B' = 4B \)[/tex]
- The height is reduced to one-third: new height [tex]\( h' = \frac{h}{3} \)[/tex]
4. Calculate the New Volume:
Substituting the new values into the volume formula:
[tex]\[
V' = \frac{1}{3} B' h' = \frac{1}{3} \times 4B \times \frac{h}{3}
\][/tex]
Simplify this expression:
[tex]\[
V' = \frac{1}{3} \times 4B \times \frac{h}{3} = \frac{4B}{3} \times \frac{h}{3} = \frac{4B h}{9}
\][/tex]
5. Compare New Volume to Original Volume:
- Original volume: [tex]\( V = \frac{1}{3} B h \)[/tex]
- New volume: [tex]\( V' = \frac{4}{9} B h \)[/tex]
6. Find the Ratio of the New Volume to the Original Volume:
The ratio of the new volume to the original volume is:
[tex]\[
\frac{V'}{V} = \frac{\frac{4}{9} B h}{\frac{1}{3} B h} = \frac{\frac{4}{9}}{\frac{1}{3}} = \frac{4}{9} \times \frac{3}{1} = \frac{4}{3} \times \frac{1}{3} = \frac{4}{3} \times \frac{1}{3} = \frac{4}{9}
\][/tex]
Thus, the correct option is:
[tex]\[
D. \ V = \frac{4}{9} B h
\][/tex]
