Chapter 9.9, Problem 31PPS

a.

To determine

To write a function to represent resulting length of box.

Answer to Problem 31PPS

  $f(x)=10-2x$

Explanation of Solution

Given information:

Open top

Length of rectangular box is 10 inches

Width of rectangular box is 8 inches

Square cut from each corner

Calculation:

Open top rectangle

Length of rectangular box in inches $=10$

Width of rectangular box $=8$

Square cut from each corner

Let x represent the side length of the square

Then resulting lenght of rectangular box in inches $=10-2x$

Function of resulting length

  $f(x)=10-2x$

b.

To determine

To write a function to represent resulting width of box.

Answer to Problem 31PPS

  $g(x)=8-2x$

Explanation of Solution

Given information:

Open top

Length of rectangular box is 10 inches

Width of rectangular box is 8 inches

Square cut from each corner

Calculation:

Open top rectangle

Length of rectangular box in inches $=10$

Width of rectangular box $=8$

Square cut from each corner

Let x represent the side length of the square cut from rectangle

Then resulting width of rectangular box in inches $=8-2x$

Function of resulting width

  $g(x)=8-2x$

c.

To determine

To find $(f\times g)(x)$ .

Answer to Problem 31PPS

  $(f\times g)(x)=80-32x+4x^2$

Explanation of Solution

Given information:

  $f(x)=10-2x$

  $g(x)=8-2x$

Calculation:

  $(f\times g)(x)=f(x)\times g(x)$

Substituting values of function of remain length and width

  $(f\times g)(x)=(10-2x)\times (8-2x)$

On solving,

  $(f\times g)(x)=80-32x+4x^2$

This indicates resulting area of rectangular box.

d.

To determine

To find domain of $(f\times g)(x)$ .

Answer to Problem 31PPS

All real numbers

Explanation of Solution

Given information:

  $(f\times g)(x)=80-32x+4x^2$

  $(f\times g)(x)=80-32x+4x^2$

Above equation is quadratic equation

And the domain of quadratic equation is always real all real numbers