Chapter P.7, Problem 46E

Solution

a.

The size of squares cut from each corner to yield a box with volume $125\text{in}{\text{.}}^3$ .

  $x\approx 0.94\text{in}\text{.},\text{ or }x\approx 3.78\text{in}\text{.}$

Given:

An open box is formed by cutting squares from each corner of a rectangular piece of cardboard as shown in image below,

  

Concept used:

• The volume of a rectangular box is given by, $\text{Volume}=\text{Length }\times \text{ Width }\times \text{ Height}$ .
• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

Let x be the side length of the squares cut from each corner.

So, after cutting the squares from each corner, the length of the resulting box will be $15-2x$ and the width would be $12-2x$ .

In order to have a box with volume 125 cubic inches, substitute these values in the volume formula and simplify further as shown below

  $\begin{array}{c}125=\left(15-2x\right)\left(12-2x\right)x\\ 125=\left(15-2x\right)\left(12x-2x^2\right)\\ 125=\left(15-2x\right)\left(12x-2x^2\right)\\ \Rightarrow 4x^3-54x^2+180x-125=0\end{array}$

To find the solutions of this cubic polynomial, plot the graph of the function $y=4x^3-54x^2+180x-125$ and locate the x -intercepts as shown below,

  

From the graph it is clear that the solutions of the equation are $x\approx 0.94,x\approx 3.78,\text{ and }x\approx 8.78$ .

Since the value of x cannot exceed 6, the solutions are $x\approx 0.94,\text{ and }x\approx 3.78$ .

Thus, the side length of the square that should be cut is either $x\approx 0.94\text{in}\text{.},\text{ or }x\approx 3.78\text{in}\text{.}$

b.

The size of squares cut from each corner to yield a box with volume more than $125\text{in}{\text{.}}^3$ .

  $0.94\text{in}\text{.}<x<3.78\text{in}\text{.}$

Given:

An open box is formed by cutting squares from each corner of a rectangular piece of cardboard as shown in image below,

  

Concept used:

• The volume of a rectangular box is given by, $\text{Volume}=\text{Length }\times \text{ Width }\times \text{ Height}$ .
• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

From the graph in above part, since the graph is above x -axis for the values between $x\approx 0.94,\text{ and }x\approx 3.78$ , the volume will be more than 125 cubic inches on the interval where x is $0.94\text{in}\text{.}<x<3.78\text{in}\text{.}$

Thus, the side length of the square that should be cut is should be between $0.94\text{in}\text{.}<x<3.78\text{in}\text{.}$

c.

The size of squares cut from each corner to yield a box with volume at most $125\text{in}{\text{.}}^3$ .

  $\left(0,0.94\right)\cup \left(3.78,6\right)$

Given:

An open box is formed by cutting squares from each corner of a rectangular piece of cardboard as shown in image below,

  

Concept used:

• The volume of a rectangular box is given by, $\text{Volume}=\text{Length }\times \text{ Width }\times \text{ Height}$ .
• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

From the graph in above part, since the graph is below x -axis for the values between $x=0\text{}\text{ to }x\approx 0.94,\text{ and from}x\approx 3.78$ to the right side.

Since the value of x cannot exceed 6, the solutions are $\left(0,0.94\right)\cup \left(3.78,6\right)$ .

Thus, the side length of the square that should be cut is in the intervals $\left(0,0.94\right)\cup \left(3.78,6\right)$