The pyramids from Part A have the same volume. See solution.
C
c
If we increase only the height or only the base area by 5, the volume of the pyramid will increase fivefold. If we increase both the height and the base area by 5, the volume will increase by the factor of 25.
Practice makes perfect
a
We are asked to draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. The area of a rectangle is equal to the product of its dimensions. For instance, 24=6* 4 and 24=12* 2.
Now, let's sketch the pyramids with the above rectangles as the bases.
\begin{gathered}
V_\text{pyramid}=\dfrac{1}{3}{\color{#0000FF}{B}}{\color{#009600}{h}}
\end{gathered}
Here, the variable B is the area of the base of the pyramid, and h is the height of the pyramid. The pyramids from Part A have the same height and their bases have the same area. Therefore, these pyramids have the same volume.
c
Let's assume that B is the base area and h is the height of a given pyramid.
\begin{gathered}
V_\text{pyramid}=\dfrac{1}{3}Bh
\end{gathered}
We are asked how multiplying the base area and/or the height by 5 affects the volume of the pyramid. Now, let's change only the height of the pyramid. Then its new height H is equal to H=5h. Let's find the volume of the new pyramid in terms of the volume of the old one.
Therefore, if we multiply only the height of the pyramid by 5 the volume of the pyramid will increase by the factor of 5. Now, let's check what would happen if we increase only the base area by 5. Then its new base area B' satisfies B'=5B.
Therefore, if we multiply only the height of the pyramid by 5, the volume of the pyramid will increase by the factor of 5. This tells us also that if we increase both the height and the base area of the pyramid by a factor of 5, we will increase the volume by a factor of 5* 5=25. Now, let's summarize the answer.
If we increase only the height or only the base area by the factor of 5, the volume of the pyramid will increase fivefold. If we increase both the height and the base area by 5, the volume will increase by the factor of 25.
