Chapter 3, Problem 21RE

a)

To determine

To express: the strength $S$ of this beam as a function of $x$ only.

Answer to Problem 21RE

  $S=13.8x\left(100-x^2\right)$

Explanation of Solution

Given information:

The strength $S$ of a wooden beam of width $x$ and depth $y$ is given by the formula $S=13.8x{y}^2$ . A beam is to be cut from a log of diameter 10 in, as shown in the figure.

  

Calculation:

Consider the formula:

  $S=13.8x{y}^2$

Here $S$ is the strength of wooden beam, $x$ is the width, and $y$ is the depth.

Diameter 10in.

Consider the diagram:

  

From the diagram,

  $\begin{array}{c}{x}^2+y^2=100\\ y^2=100-x^2\end{array}$

To express the strength $S$ of the beam as a function of $x$ only, use $y^2=100-x^2$ :

Use $y^2=100-x^2$ in the formula $S=13.8x{y}^2$ ;

  $S=13.8x\left(100-x^2\right)$

b)

To determine

To find: the domain of the function $S$ .

Answer to Problem 21RE

Domain of the unction is set of real numbers.

Explanation of Solution

The strength $S$ of a wooden beam of width $x$ and depth $y$ is given by the formula $S=13.8x{y}^2$ . A beam is to be cut from a log of diameter 10 in, as shown in the figure.

  

The function for strength is $S=13.8x\left(100-x^2\right)$ .

Domain of the function is all real numbers.

c)

To determine

To draw: a graph of $S$ .

Explanation of Solution

Given information:

The strength $S$ of a wooden beam of width $x$ and depth $y$ is given by the formula $S=13.8x{y}^2$ . A beam is to be cut from a log of diameter 10 in, as shown in the figure.

  

Calculation:

To draw the graph of the function $S=13.8x\left(100-x^2\right)$ ;

  

d)

To determine

To find: the width that make the beam strongest.

Answer to Problem 21RE

The beam strongest, the width will be 5.8in.

Explanation of Solution

Given information:

The strength $S$ of a wooden beam of width $x$ and depth $y$ is given by the formula $S=13.8x{y}^2$ . A beam is to be cut from a log of diameter 10 in, as shown in the figure.

  

Calculation:

To make the beam strongest, use derivative with respect to $x$ for the function,

  $\begin{array}{c}S=13.8x\left(100-x^2\right)\\ =1380x-13.8x^3\end{array}$

Differentiate with respect to $x$ ;

  $\frac{dS}{dx}=1380-41.4x^2$

To make the beam strongest, the derivative must be equal to zero, i.e, $\frac{dS}{dx}=0$ ;

  $\begin{array}{c}1380-41.4x^2=0\\ 41.4x^2=1380\\ x^2=33.3333\\ x\approx 5.8\end{array}$

Therefore, to make the beam strongest, the width will be 5.8in.