Question

The dimensions of a cuboid are in the ratio $5:2:1$. It's volume is $1250$ cubic meters. Find the total surface area.

Answer 1

Step 1. Determine the dimensions of the cuboid :

Assume the common ratio be $x$. So, the dimensions will be $5x,2x\mathrm{and}x$. The volume of the cuboid is the product of the dimensions of the cuboid,

$\begin{array}{rcl}\mathrm{Volume}& =& 5x\times 2x\times x\\ & \Rightarrow & 1250=10x^3\\ & \Rightarrow & 125=x^3\\ & \Rightarrow & 5=x\end{array}$

Determine the dimensions of the cuboid,

$\begin{array}{rcl}5x& =& 5\times 5=25\\ 2x& =& 2\times 5=10\\ x& =& 5\end{array}$

So, the dimension of the cuboid are $25\mathrm{m},10\mathrm{m}\mathrm{and}5\mathrm{m}$.

Step 2. Determine the total surface area of the cuboid :

The total surface area includes the top part, base part, and the curve part. Determine the total surface area of the cuboid,

$\begin{array}{rcl}\mathrm{TSA}& =& 2\left(25\times 10+10\times 5+5\times 25\right)\\ & \Rightarrow & \mathrm{TSA}=2\left(250+50+125\right)\\ & \Rightarrow & \mathrm{TSA}=2\left(425\right)\\ & \Rightarrow & \mathrm{TSA}=850\end{array}$

Hence, the total surface area of the cuboid is $850{\mathrm{m}}^2$.