Chapter 7.2, Problem 46IP

(a)

To determine

A table to show the bacteria population for the first 4 hours

Answer to Problem 46IP

Width of a Rectangle	Perimeter   $P=2\left(l+w\right)$
1	6
2	12
3	18
4	24
5	30
6	36

Explanation of Solution

Given:

A rectangle that has a length that is twice its width

  $\begin{array}{l}width=x\\ length=2x\end{array}$

Concept Used:

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

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Calculation:

In order to make a table that shows the width of a rectangle and its perimeter for widths of 1, 2,3,4,5, and 6 unit is below:

  $\begin{array}{c}Width=1\\ Perimeter=2\left(x+2x\right)\\ =2\left(1+2\cdot 1\right)\\ =6\\ Width=2\\ Perimeter=2\left(x+2x\right)\\ =2\left(2+2\cdot 2\right)\\ =12\\ Width=3\\ Perimeter=2\left(x+2x\right)\\ =2\left(3+2\cdot 3\right)\\ =18\\ Width=4\\ Perimeter=2\left(x+2x\right)\\ =2\left(4+2\cdot 4\right)\\ =24\\ Width=5\\ Perimeter=2\left(x+2x\right)\\ =2\left(5+2\cdot 5\right)\\ =30\\ Width=6\\ Perimeter=2\left(x+2x\right)\\ =2\left(6+2\cdot 6\right)\\ =36\end{array}$

Thus, the table is below:

Width of a Rectangle	Perimeter   $P=2\left(l+w\right)$
1	6
2	12
3	18
4	24
5	30
6	36

Thus, the table is above for the perimeter of rectangle

(b)

To determine

Graph the ordered pairs (width, perimeter)

Answer to Problem 46IP

The Graph

Explanation of Solution

Given: the ordered pairs are $\left(1,6\right),\left(2,12\right),\left(3,18\right),\left(4,24\right),\left(5,30\right),\left(6,36\right)$

Width of a Rectangle	Perimeter   $P=2\left(l+w\right)$
1	6
2	12
3	18
4	24
5	30
6	36

Concept Used:

• In rectangular coordinate system any point be in the form of $\left(x,y\right)$ .
• In Quadrant I, both the coordinates of ordered pair are positive, that is, $\left(+,+\right)$ .
• In Quadrant II, first number is negative and second is positive, that is, $\left(-,+\right)$ .
• In Quadrant III, both the coordinates of ordered pair are negative, that is, $\left(-,-\right)$ .
• In Quadrant IV, first point is positive and second is negative, that is, $\left(+,-\right)$ .

In order to plot the point $\left(1,6\right),\left(2,12\right),\left(3,18\right),\left(4,24\right),\left(5,30\right),\left(6,36\right)$ , observe that both numbers are same they are positive, so the point will be in first quadrant.

To plot the graph of the point $\left(1,6\right)$ , first move 1 units to the right starting from origin and then move 6 units up.

To plot the graph of the point $\left(2,12\right)$ , first move 2 units to the right starting from origin and then move 12 units up.

To plot the graph of the point $\left(3,18\right)$ , first move 3 units to the right starting from origin and then move 18 units up.

To plot the graph of the point $\left(4,24\right)$ , first move 4 units to the right starting from origin and then move 24 units up.

To plot the graph of the point $\left(5,30\right)$ , first move 5 units to the right starting from origin and then move 30 units up.

To plot the graph of the point $\left(6,36\right)$ , first move 6 units to the right starting from origin and then move 36 units up.

The graph of the point is shown below:

(c)

To determine

An expression in simplest form for the perimeter of the rectangle

Answer to Problem 46IP

The expression is $P=6x$ .

Explanation of Solution

Given:

A rectangle that has a length that is twice its width

  $\begin{array}{l}width=x\\ length=2x\end{array}$

Concept Used:

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

Calculation:

In order to find the expression for the perimeter is below:

  $\begin{array}{c}P=2\left(length+width\right)\\ P=2\left(2x+x\right)\\ =2\left(3x\right)\\ P=6x\end{array}$

Thus, the expression is $P=6x$ .