Chapter 11.7, Problem 1MP

a.

To determine

The equation of a ration function that gives the surface area of the lunch box in terms of x .

Answer to Problem 1MP

  $S\left(x\right)=\frac{2x^3+1420}{x}$

Explanation of Solution

Given:

The volume of the lunch box is, $355\text{in}{\text{.}}^3$

  

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
1. Multiply both sides of equations by LCD(least common denominator) of all denominators.
2. Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

The expression for the surface area of the lunch box is given by,

  $S=2x^2+\frac{1420}{x}$

So, in the expression of surface area, the first term has no denominator, so it can be considered 1 as its denominator. The denominator of second term is x , so the least common denominator of 1 and x should be x . So, multiply the first term by x up and down and then simplify the numerators over the common denominator as shown below,

  $\begin{array}{l}S=2x^2\cdot \frac{x}{x}+\frac{1420}{x}\\ S=\frac{2x^3}{x}+\frac{1420}{x}\\ S=\frac{2x^3+1420}{x}\end{array}$

Thus, the equation of a rational expression for the surface area of lunch box is $S\left(x\right)=\frac{2x^3+1420}{x}$ .

b.

To determine

The graph of the function for $x>0$ .

Explanation of Solution

Given:

The volume of the lunch box is, $355\text{in}{\text{.}}^3$

  

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
3. Multiply both sides of equations by LCD(least common denominator) of all denominators.
4. Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

The graph of the surface area function is shown below,

  

The graph represents the amount of cardboard needed S(x) on vertical axis, for the side length x of the box in horizontal axis.

c.

To determine

The coordinate of the minimum point of the function.

Answer to Problem 1MP

  $\left(7.08,300.82\right)$

Explanation of Solution

Given:

The volume of the lunch box is, $355\text{in}{\text{.}}^3$

  

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
5. Multiply both sides of equations by LCD(least common denominator) of all denominators.
6. Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

The graph of the surface area function is shown below,

  

The graph represents the amount of cardboard needed S(x) on vertical axis, for the side length x of the box in horizontal axis.

From the graph it is clear the coordinates of the minimum point rounded to nearest hundredth is $\left(7.08,300.82\right)$ .

d.

To determine

What does the minimum point tell.

Explanation of Solution

Given:

The volume of the lunch box is, $355\text{in}{\text{.}}^3$

  

Calculation:

The minimum point of the graph tells that when the length of the base of the box is $7.08\text{ in}\text{.}$ , the surface area of the minimum cardboard needed to make the required lunch box is $300.82\text{in}{\text{.}}^2$ .