Chapter 12.5, Problem 2AE

To determine

Tofind:The value of $x$ for which the greatest value of area occurs.

Answer to Problem 2AE

The value of $x$ for which the greatest value of area occurs is $2$ .

Explanation of Solution

Given information:

The original triangle had sides $5$ , $5$ , $8$ instead of $5$ , $5$ , $6$ and the area is equal to $\text{A}=\frac{3x\left(4-x\right)}{2}$ .

Calculation:

Check the maximum value of the area function and the point of maximum value with the help of graph.

To graph a function $\text{A}=\frac{3x\left(4-x\right)}{2}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{3x\left(4-x\right)}{2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{3}\boxed{\text{X}}\boxed{(}\boxed{4}\boxed{-}\boxed{\text{X}}\boxed{)}\boxed{)}\boxed{÷}\boxed{2}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(3\text{X}\left(4-\text{X}\right)\right)/2$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from the graph, the maximum area of the rectangle occurs at $x=2$ and is equal to $6$ .

Therefore, the value of $x$ for which the greatest value of area occurs is equal to $2$ .