Answer

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**Hint:**In this problem we have the cost to make a metal box and the cost of the sheet per square meter. They have asked to calculate one of the dimensions of the box which is height. For this we will first calculate the surface area of the box by dividing the cost to make a metal box by the cost of a sheet per square meter. Now we will assume the height of the box as $h$ and we will calculate the surface area of the box by using the formula $2\left( lb+bh+lh \right)$. From this formula we will calculate the surface area of the box in terms of $h$ and we will equate it to the calculated surface area from the costs given. Now we will simplify the obtained equation to get the required result.

**Complete step by step solution:**

Given that, A box made of sheet metal costs Rs.$6480$ at Rs. $120$ per square meter.

Now the surface area of the box will be

$\begin{align}

& s=\dfrac{6480}{120} \\

& \Rightarrow s=54{{m}^{2}} \\

\end{align}$

In the problem we have the dimensions of the box as

Length $l=5m$,

Width $b=3m$,

Let the height of the box is $h$.

Now the surface area of the box is given by

$\Rightarrow s=2\left( lb+bh+lh \right)$

Substituting the values, we have in the above equation, then we will get

$\begin{align}

& \Rightarrow s=2\left( 5\times 3+3\times h+5\times h \right) \\

& \Rightarrow s=2\left( 15+3h+5h \right) \\

& \Rightarrow s=30+16h \\

\end{align}$

But we have the surface area as $s=54{{m}^{2}}$, equating the both the values, then we will have

$\Rightarrow 30+16h=54$

Simplifying the above equation, then we will get

$\begin{align}

& \Rightarrow 16h=54-30 \\

& \Rightarrow h=\dfrac{24}{16} \\

& \Rightarrow h=1.5m \\

\end{align}$

**Hence the height of the metal box is $1.5$m and the box are look like below**

**Note:**When we are dealing with the three dimensional shaped, we need to understand the problem clearly and know what value we need to calculate. When they have mentioned painting, metal required, then we need to calculate the surface area. If they have mentioned that one shape is casted into another shape then we will consider the volume of the shapes.

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