$\"\"$

\

The equations are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

\

The length of the beam is $\"\"$ .

\

The volume of the beam is $\"\"$ .

\

The area of the beam is $\"\"$ .

\

Substitute $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Substitute $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

If $\"\"$ and $\"\"$ .

\

Substitute $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Substitute $\"\"$ and $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Therefore, the volume of the beam is $\"\"$

\

$\"\"$

\

Find the weight of the beam.

\

The density of the concrete is $\"\"$

\

The weight of the beam is $\"\"$ .

\

Substitute corresponding values.

\

$\"\"$

\

$\"\"$

\

Therefore, the weight of the beam is $\"\"$

\

$\"\"$

\

(b)

\

Find the centroid of the cross section of the beam.

\

The centroid of the beam is $\"\"$ .

\

The beam is symmetric about $\"\"$ -axis, hence the value of the $\"\"$ .

\

The value of $\"\"$ .

\

Substitute $\"\"$ .

\

$\"\"$ .

\

The value of $\"\"$

\

$\"\"$ .

\

Apply formula : $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Substitute $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Therefore, the centroid of the beam is $\"\"$ .

\

$\"\"$

\

(a) The volume of the beam is $\"\"$ and the weight of the beam is $\"\"$

\

(b) The centroid of the beam is $\"\"$ .