3. A regular quadrilateral prism has a surface area of $2520 \mathrm{~cm}^{2}$. If this prism were $3 \mathrm{~cm}$ shorter, its surface area would be $2304 \mathrm{~cm}^{2}$. What would its volume be in that case?

(8 points)
3. A regular quadrilateral prism has a square base with an area of $S=a^{2}$ and a perimeter of $o=4 a$. Since the prism is regular, it is also a right prism, and its surface area is calculated as $P=2 S+o \cdot v$. The difference in surface areas in both cases is then $P-P_{2}=2 S+o \cdot v-\left(2 S+o \cdot v_{2}\right)=o \cdot v-o \cdot v_{2}=$ $o\left(v-v_{2}\right)$. Substituting the given data, we get $2520-2304=4 a \cdot 3$, or $216=12 a$ and $a=18 \mathrm{~cm}$. If we now substitute this into the formula for the surface area of one of the prisms (e.g., the original one), we get $P=2 a^{2}+4 a \cdot v$ and $2520=2 \cdot 18^{2}+4 \cdot 18 \cdot v$. From this, we calculate that the height of the original prism is $26 \mathrm{~cm}$, and the height of the second prism is $23 \mathrm{~cm}$. Considering the formula for calculating the volume of the prism $V=S \cdot v=18^{2} \cdot 23=7452 \mathrm{~cm}^{3}$.

Noting or considering that the base is a square with an area

$S=a^{2}$ and a perimeter of $o=4 a$. 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-10.jpg?height=51&width=1642&top_left_y=1228&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-10.jpg?height=52&width=1642&top_left_y=1276&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-10.jpg?height=57&width=1642&top_left_y=1325&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-10.jpg?height=52&width=1636&top_left_y=1376&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-10.jpg?height=55&width=1642&top_left_y=1423&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-10.jpg?height=48&width=1642&top_left_y=1478&top_left_x=204)

Calculating the volume $V=S \cdot v=18^{2} \cdot 23=7452 \mathrm{~cm}^{3} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point