Chapter 9.8, Problem 23PPS

(a)

To determine

To calculate: To write equations to represent the perimeter $P$ and the area $A$ of the square in term of its side lengths.

Answer to Problem 23PPS

Function to represent the perimeter of square is $P=4s$

Function to represent the area of square is $A=s^2$

Explanation of Solution

Given information: A square has a side length $s$

Formula Used:

Perimeter of square of side length is given as

  $Perimeter=4\times Side$

Area of square of side length is given as

  $Area=Sid{e}^2$

Calculation:

Given a square has a side length $s$

Function to represent the perimeter of square is given as

  $\begin{array}{l}Perimeter=4\times Side\\ P=4\times s\\ P=4s\end{array}$

Function to represent the area of square is given as

  $\begin{array}{l}Area=Sid{e}^2\\ A=s^2\end{array}$

Conclusion:

Hence, function to represent the perimeter of square is $P=4s$

Function to represent the area of square is $A=s^2$

(b)

To determine

To calculate: To make a table showing the perimeter and area of the square for the side length $0,1,2,3,4$ units

Answer to Problem 23PPS

Table is

Side Length	Perimeter	Area
0	0	0
1	4	1
2	8	4
3	12	9
4	16	16

Explanation of Solution

Given information: A square has a side length $s$

Calculation:

Given a square has a side length $s$

Function to represent the perimeter of square is $P=4s$

Function to represent the area of square is $A=s^2$

When $s=0$

  $\begin{array}{l}P=4\left(0\right)=0\\ A={\left(0\right)}^2=0\end{array}$

When $s=1$

  $\begin{array}{l}P=4\left(1\right)=4\\ A={\left(1\right)}^2=1\end{array}$

When $s=2$

  $\begin{array}{l}P=4\left(2\right)=8\\ A={\left(2\right)}^2=4\end{array}$

When $s=3$

  $\begin{array}{l}P=4\left(3\right)=12\\ A={\left(3\right)}^2=9\end{array}$

When $s=4$

  $\begin{array}{l}P=4\left(4\right)=16\\ A={\left(4\right)}^2=16\end{array}$

Thus, table showing the perimeter and area of square with different side lengths is:

Side Length	Perimeter	Area
0	0	0
1	4	1
2	8	4
3	12	9
4	16	16

Conclusion:

Hence, table is created

(c)

To determine

To calculate: To graph the points whose ordered pairs are (side length, perimeter) and (side length, area) and describe the graphs

Answer to Problem 23PPS

Graph of ordered pair (side length, perimeter) is a straight line

Graph of ordered pair (side length, area) is an upward parabola

Explanation of Solution

Given information: A square has a side length $s$

Calculation:

Given a square has a side length $s$

Function to represent the perimeter of square is $P=4s$

Function to represent the area of square is $A=s^2$

Thus, table showing the perimeter and area of square with different side lengths is:

Side Length	Perimeter	Area
0	0	0
1	4	1
2	8	4
3	12	9
4	16	16

Ordered pairs (side length, perimeter) is given as

  $\left(0,0\right),\left(1,4\right),\left(2,8\right),\left(3,12\right),\left(4,16\right)$

Plotting the ordered pairs (side length, perimeter) on graph:

  

Graph is a straight line

Ordered pairs (side length, area) is given as

  $\left(0,0\right),\left(1,4\right),\left(2,8\right),\left(3,12\right),\left(4,16\right)$

Plotting the ordered pairs (side length, area) on graph:

  

Graph is an upward parabola

Conclusion:

Hence, graphs are sketched

Graph of ordered pair (side length, perimeter) is a straight line

Graph of ordered pair (side length, area) is an upward parabola

(d)

To determine

To calculate: To find if there are any values for $s$ that have the same numerical value for $A$ and $P$

Answer to Problem 23PPS

Yes, the values of $A$ and $P$ are same for $s=4$

Explanation of Solution

Given information: A square has a side length $s$

Calculation:

Given a square has a side length $s$

Function to represent the perimeter of square is $P=4s$

Function to represent the area of square is $A=s^2$

Thus, table showing the perimeter and area of square with different side lengths is:

Side Length	Perimeter	Area
0	0	0
1	4	1
2	8	4
3	12	9
4	16	16

From the table, numerical value of $A$ and $P$ are same for $s=4$

Conclusion:

Hence, $A$ and $P$ have same value for $s=4$