Chapter 4.1, Problem 84E

To determine

To find all possible dimensions of the rectangle. Also the dimension with highest perimeter.

Answer to Problem 84E

  $x=-2$

Explanation of Solution

Given Information: Area of rectangle= 54sq. inches.

Formula used:

Area of rectangle= $length\times breath$

Perimeter of rectangle= $2\left(length+breath\right)$

Calculation:

To find all possible dimensions of the rectangle, must find all the factors of the area and then find out the factor pair of it.

Since the Factors of 54 are all the numbers that you can evenly divide into 54, simply need to divide 54 by all numbers up to 54 to see which ones result in an even quotient.

  $\begin{array}{l}54÷1=54\\ 54÷2=27\\ 54÷3=18\\ 54÷6=9\\ 54÷9=6\\ 54÷18=3\\ 54÷27=2\\ 54÷54=1\end{array}$

The Positive Factors of 54 are therefore all the numbers used to divide (divisors) above to get an even number. Here is the list of all Positive Factors of 54 in numerical order:

1, 2, 3, 6, 9, 18, 27, and 54.

Factor Pairs of 54

Factor Pairs of 54 are combinations of two factors that when multiplied together equal 54. Here are all the Positive Factor Pairs of 54

  $\begin{array}{l}1\times 54=54\\ 2\times 27=54\\ 3\times 18=54\\ 6\times 9=54\\ 9\times 6=54\\ 18\times 3=54\\ 27\times 2=54\\ 54\times 1=54\end{array}$

These factor pairs are the list of all possible dimensions that the rectangle of area 54 square inches.

Now to find the dimension with largest perimeter, need the formula of perimeter of rectangle i.e. Perimeter of rectangle= $2\left(length+breath\right)$

If put the dimension $1\times 54$ in formula, will have the largest perimeter.

  $\begin{array}{l}Perimeterofrectangle=2\left(length+breath\right)\\ PM=2\left(1+54\right)\\ PM=2(55)\\ PM=110inches\end{array}$