Question

The height of a right square pyramid is multiplied by $10$, but the area of the base is unchanged. By what factor is the volume of the pyramid multiplied?

Answer 1

Find the factor of change in volume:

We know that, the volume of right square pyramid of base area $A$ and height $h$ is given by $V=\frac{Ah}{3}$.

If the height of the pyramid is multiplied by $10$, then new height is given by $h\text{'}=10h$.

The base area of the pyramid is unchanged, so $A\text{'}=A$.

Now the volume of new pyramid will be $V\text{'}=\frac{A\text{'}h\text{'}}{3}$.

Substitute $h\text{'}=10h$ and $A\text{'}=A$ in $V\text{'}=\frac{A\text{'}h\text{'}}{3}$.

$\begin{array}{rcl}V\text{'}& =& \frac{A·10h}{3}\\ & \Rightarrow & V\text{'}=10\left(\frac{Ah}{3}\right)\\ \therefore V\text{'}& =& 10V\end{array}$

Therefore, new volume will be $10$ times multiple of original volume.

Hence, the volume of the pyramid multiplied by $10$.