Chapter 7.2, Problem 14P

To determine

Find the dimensions of the original piece of metal.

Answer to Problem 14P

The dimensions of the original piece of metal = 14.47 cm (each side)

Explanation of Solution

Given:

A box with a square base and no lid is to be made from a square piece of metal by cutting squares from the corners and folding up the sides . The cut-off squares are 5 cm on a side.The volume of the box is 100 cubic cm .

Calculation :

We know that the base of the box is a square. Let its sides be x cm each .

The cut outs are squares with sides 5 cm.

Since the metal sheet is also square and the cutouts are squares, so the remaining sides of the figure have measures equal to opposite sides.

So, we draw the diagram :

  

Now, since the volume of a box is the multiplication of base area and the height of the box.

Since the base of the box is a square with sides = x cm .

Area of the base = $x^2$

When you fold the pieces upward to form the box , you find that the height of the box is 5 cm.

So, the volume of the box = $x^2\cdot 5$

Given , volume of the box = 100 cubic cm

Equate both and solve for x :

  $\begin{array}{l}5x^2=100\\ \Rightarrow x^2=\frac{100}{5}\mathrm{........}[\text{divide each side by 5]}\\ \Rightarrow x^2=20\\ \Rightarrow x=\sqrt{20}\mathrm{.......}[\text{take square root each side]}\\ \Rightarrow x=2\sqrt{5}\text{ = 4}\text{.47 cm}\end{array}$

The original piece of metal is a square with each side = 5 + 5 + x = 10 + x = 14.47 cm

So, the dimensions of the original piece of metal = 14.47 cm (each side)