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

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Solution

Step 1. Determine the dimensions of the cuboid :
Assume the common ratio be $x$ . So, the dimensions will be $5 x, 2 x and x$ . The volume of the cuboid is the product of the dimensions of the cuboid,
$\begin{array}{rcl} Volume & = & 5 x \times 2 x \times x \\ \Rightarrow & 1250 = 10 x^{3} \\ \Rightarrow & 125 = x^{3} \\ \Rightarrow & 5 = x \end{array}$
Determine the dimensions of the cuboid,
$\begin{array}{rcl} 5 x & = & 5 \times 5 = 25 \\ 2 x & = & 2 \times 5 = 10 \\ x & = & 5 \end{array}$
So, the dimension of the cuboid are $25 m, 10 m and 5 m$ .
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,
$\begin{array}{rcl} TSA & = & 2 (25 \times 10 + 10 \times 5 + 5 \times 25) \\ \Rightarrow & TSA = 2 (250 + 50 + 125) \\ \Rightarrow & TSA = 2 (425) \\ \Rightarrow & TSA = 850 \end{array}$
Hence, the total surface area of the cuboid is $850 m^{2}$ .

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