Q. 32 - Precalculus Enhanced with Graphing Utilities

Question

Cross-sectional Area The cross-sectional area of a beam

cut from a log with radius 1 foot is given by the function

$A\left(x\right)=4x\sqrt{1-x^2}$, where x represents the length, in feet,

of half the base of the beam. See the figure.

(a) Find the domain of A.

(b) Use a graphing utility to graph the function A = A(x).

(c) Create a TABLE with TblStart = 0 and Tbl = 0.1. 

Which value of x in the domain found in part (a) maximizes the cross sectional area? What should be the length of the base of the beam to maximize the cross-sectional area?

Step-by-Step Solution

Answer

(a).  The Domain of the function is -  $\left\{x\right|-1\le x\le 1\}$

(b).  

(c). 

$x$	$A\left(x\right)=4x\sqrt{1-x^2}$
0	0
0.1	0.39799497
0.2	0.78383672
0.3	1.144727
0.4	1.4664242
0.5	1.7320508
0.6	1.92
0.7	1.9996
0.8	1.92
0.9	1.5692036
1	0

$$\begin{gathered} x=0.7 \\ length = 0.14 \end{gathered}$$

Part (a). Step 1. Given Data

Function -  $A\left(x\right)=4x\sqrt{1-x^2}$

Part (a). Step 2. To Find

Find the domain of A.

Part (a). Step 3. Explanation

$A\left(x\right)=4x\sqrt{1-x^2}$

The Function of A tells us to takes the square roots of $1-x^2$ Since only the nonnegative number have real square roots, cannot be negative, Therefore we can write -

$1-x^2\ge 0$

Add ${\mathit{x}}^{\mathbf{2}}$ on both side -

$1\ge x^2$

This can be written as -

$x^2\le 1$

The above inequality is only true when   $-1\le x\le 1$

The Domain of function is    $\{\overline{)x}-1\le x\le 1\}$

Part (b). Step 1. Given Data

Function -   $A\left(x\right)=4x\sqrt{1-x^2}$

Part (b). Step 2. To Find

Use a graphing utility to graph the function A = A(x).

Part (b). Step 3. Explanation

Graph the function using graphing utility -

Part (c). Step 1. Given Data

Function -  $A\left(x\right)=4x\sqrt{1-x^2}$

Part (c). Step 2. To Find

Create a TABLE with TblStart = 0 and Tbl = 0.1. Which value of x in the domain found in part (a) maximizes the cross-sectional area? What should be the length of the base of the beam to maximize the cross - sectional area?

Part (c). Step 3. Explanation

Using the graphing utility, Create the table as stated in the book based on the table, the greatest area is when x is 0.7 since x represents half the base of the beam, then the length should be 0.14.

$x$	$4x\sqrt{1-x^2}$
0	0
0.1	0.39799497
0.2	0.78383672
0.3	1.144727
0.4	1.4664242
0.5	1.7320508
0.6	1.92
0.7	1.9996
0.8	1.92
0.9	1.5692036
1	0

$$\begin{gathered} x=0.7 \\ length = 0.14 \end{gathered}$$