Question

Suppose that you are the manager of a sheet metal shop. A customer asks you to manufacture $10,000$boxes, each box being open on top. The boxes are required to have a square base and a $9$-cubic-foot capacity. You construct the boxes by cutting out a square from each corner of a square piece of sheet metal and folding along the edges.

Part (a): What are the dimensions of the square to be cut if the area of the square piece of sheet metal is $100$square feet?

Part (b): Could you make the box using a smaller piece of sheet metal? Make a list of the dimensions of the box for various pieces of sheet metal.

Short answer

Part (a): The dimension is $4.274ftby4.274ft$or $0.093ftby0.093ft$.

Part (b): Side of the square base of the box may vary as $1.45<y<9.81$and for these values of y. We can get different values of x by the equation (i), which is the height of the box.

Step-by-step solution

Part (a) Step 1. Given information.

Consider the given question,

No. of boxes$=10,000$

Volume of the boxlocalid="1647016433327" $=9f{t}^3$

Area of metal sheetlocalid="1647016437319" $=100f{t}^2$

Assume the side of the base of the box is y and height of the box is x.

Area of metal sheetlocalid="1647016442831" $$\begin{gathered} =\sqrt{100} \\ =10ft \end{gathered}$$

From the question,

$$\begin{gathered} y^{2}x=9......\left(i\right) \\ y+2x=10 \\ y=10-2x......\left(ii\right) \end{gathered}$$

Part (a) Step 2. Substitute the value of y in equation (i).

Substitute the value of y in equation (i),

$$\begin{gathered} {\left(10-2x\right)}^{2}x=9 \\ 4x^3-40x^2+100x-9=0 \end{gathered}$$

Plot the function,

Part (a) Step 3. Use the graph and solve the equation.

Consider the above graph,

$x=0.093,4.274,5.632$

From equation (ii), we see that if $x>5$ then y becomes negative, which is not possible. So, $x=5.632$is not acceptable.

Therefore, the dimensions of the square to be cut are$4.274ftby4.274ft$or$0.093ftby0.093ft$.

Part (b) Step 1. Given information.

Consider the given question,

No. of boxes$=10,000$

Volume of the box$=9f{t}^3$

Assume the side of the base of the box is y and height of the box is x.

From the question,

$$\begin{gathered} y^{2}x=9......\left(i\right) \\ x=\frac{9}{y^2} \\ y+2x<10......\left(ii\right) \end{gathered}$$

Part (b) Step 2. Substitute the value of x in equation (ii).

Substitute the value of x in equation (ii),

$$\begin{gathered} y+2\left(\frac{9}{y^2}\right)<10 \\ {y}^3+18-10y^2<0 \end{gathered}$$

Plot the function,

Part (b) Step 2. Find the value of y.

Consider the above graph,

We see that $f\left(y\right)<0$for $1.45<y<9.81$

Therefore, we can make the box using a smaller piece of sheet metal. And the side of the square base of the box may vary as $1.45<y<9.81$ and for these values of y. We can get different values of x by the equation (i), which is the height of the box.