Chapter 5.5, Problem 39E

(a)

To determine

To find: the value at which perimeter and area of triangle equal to each other using graph.

Answer to Problem 39E

  $x=10,y=24.$

Explanation of Solution

Given:

  

Concept used:

Perimeter of the triangle is:

Sum of the all side of the triangle.

Area of the triangle is:

  $\frac{1}{2}\times \text{base}\times \text{height}$ .

Calculation:

The objective is to write and solve the system for the value of $x$ .

The perimeter of the triangle equals to area of the triangle.

Sketch the following diagram:

  

The perimeter of triangle is:

  $\begin{array}{l}P=x+6+\left(x-2\right).\\ P=2x+4.\end{array}$

Area of triangle is:

  $\begin{array}{l}A=\frac{1}{2}.\left(x-2\right)\mathrm{.6.}\\ \Rightarrow 3x-6.\end{array}$

Equate the perimeter to area of the triangle:

  $2x+4=3x-6.$

Step1. Write a system of linear equations using each side of the original equation,

Let,

  $\begin{array}{l}y=2x+4.\\ y=3x-6.\end{array}$

Construct the following table:

$X$	$Y_1$	$Y_2$
$2$	$8$	$0$
$4$	$12$	$6$
$6$	$16$	$12$
$8$	$20$	$18$
$10$	$24$	$24$
$12$	$28$	$30$

Hence, at $x=10,y=24.$

(b)

To determine

To find: The Domain of the function.

Answer to Problem 39E

Domain of the function is $\left\{x\right|x\in R\}$ .

Explanation of Solution

Given: ${\text{g(x)=8}}^{\text{x}}$

Concept used:

The power of the any number must be real number.

Calculation:

  

It is clear from the graph both the lines intersecting at $x=10.$

Hence, the result is $x=10.$