Chapter 8.3, Problem 30E

a.

To determine

To write:

An equation describing the possible values of x and y .

Answer to Problem 30E

The equation $2x+2y=16$ represents the possible values of x and y .

Explanation of Solution

Given:

The rectangle shown has a perimeter of 16 inches.

  

Calculation:

We know that the perimeter of a rectangle is sum of two times length and two times width. Upon representing our given information in an equation, we will get:

  $2x+2y=16$

Therefore, the equation $2x+2y=16$ represents the possible values of x and y .

b.

To determine

To graph:

The equation from part (a) using intercepts.

Answer to Problem 30E

x -intercept: $(8,0)$

y -intercept: $(0,8)$

Explanation of Solution

Given:

The rectangle shown has a perimeter of 16 inches.

  

Calculation:

To find x -intercept, we will substitute $y=0$ in our given equation and solve for xas shown below:

  $\begin{array}{l}2x+2y=16\left(\text{Given equation}\right)\\ 2x+2(0)=16\left(\text{Substituting }y=0\right)\\ 2x=16\\ \frac{2x}{2}=\frac{16}{2}\left(\text{Dividing both sides by 2}\right)\\ x=8\end{array}$

Therefore, the x -intercept of our given equation is $(8,0)$ .

To find y- intercept, we will substitute $x=0$ in our given equation and solve for yas shown below:

  $\begin{array}{l}2x+2y=16\left(\text{Given equation}\right)\\ 2(0)+2y=16\left(\text{Substituting }x=0\right)\\ 2y=16\\ \frac{2y}{2}=\frac{16}{2}\left(\text{Dividing both sides by 2}\right)\\ y=8\end{array}$

Therefore, the y -intercept of our given equation is $(0,8)$ .

Upon graphing our given equation, we will get our required graph as shown below:

  

c.

To determine

To give:

Three pairs of whole number values of x and y that could represent side lengths of rectangle.

Answer to Problem 30E

Our required pairs would be $(1,7),(3,5)\text{}\text{ and (5,3)}$ .

Explanation of Solution

Given:

The rectangle shown has a perimeter of 16 inches.

  

Calculation:

To get three possible pairs for side lengths, we need to identify 3 points on the line graphed in part (b).

We can see that the points $(1,7),(3,5)\text{}\text{ and (5,3)}$ lie on our line.

Therefore, the points $(1,7),(3,5)\text{}\text{ and (5,3)}$ represent three pairs for values of x and y that could represent side lengths of rectangle.

d.

To determine

To explain:

Does either the x -intercept or the y -intercept represent a possible side length of the rectangle.

Answer to Problem 30E

Neither x -intercept nor the y -intercept represent a possible side length of the rectangle.

Explanation of Solution

Given:

The rectangle shown has a perimeter of 16 inches.

  

Calculation:

We know that the x -intercept is the point on x -axis, where the value of y is 0. The y -intercept is the point on y -axis, where the value of x is 0.

We also know that to form a rectangle we need two sides and both sides should be greater than 0. At the both intercepts, one side will be 0.

Therefore, neither x -intercept nor the y -intercept represent a possible side length of the rectangle.