The dimension of a cuboid area in the ratio 4:5:6 and the total surface area is 5,328cm². Find the volume and the cost of painting inner surface area without the upper face, if the rate of painting is Rs. 25 per square meter.

The dimension of cuboid are in the ratio 4 : 5 : 6.
So, suppose the length, width and height of cylinder are 4x , 5x and 6x respectively.
Then its total surface area = $2\left(lw+wh+lh\right) = 2\left(4x\times 5x+5x\times 6x+4x\times 6x\right) = 2\times 74x^2 = 148x^2$
And total surface area given is 5328 sq. cm. So we have;
$$\begin{gathered} 148x^2 = 5328 \\ \Rightarrow x^2 = \frac{5328}{148} = 36 \\ \Rightarrow x = \sqrt{36} = 6 \end{gathered}$$
So, length of the cuboid = $4\times 6 = 24 \mathrm{cm}$
Width of cuboid = $5\times 6 = 30 \mathrm{cm}$
Height of cuboid = $6\times 6 = 36 \mathrm{cm}$
So volume of cuboid = $lwh = 24\times 30\times 36 = 25920 {\mathrm{cm}}^3$
Inner surface area without the upper face = lateral surface area + area of base = 2h(l + w) + lw
$$\begin{gathered} = 2\times 36\left(24+30\right)+24\times 30 \\ = 2\times 36\times 54+720 \\ = 4608 {\mathrm{cm}}^2 \end{gathered}$$
So total area to be painted = $4608 {\mathrm{cm}}^2 = \frac{4608}{10000} {\mathrm{m}}^2 = 0.4608 {\mathrm{m}}^2$
Cost of painting per square metre = Rs.25
Therefore total cost of painting the inner surface without upper face = $25\times 0.4608$= Rs.11.52