The diagrams in Fig. 15.9 show the cross-sections of steel beams. All dimensions are in cm. Calculate the volumes, in
, of 5-metre lengths of the beams.
The Correct Answer and Explanation is:
Here are the calculated volumes for the 5-metre lengths of each steel beam.
- Volume of beam a = 5000 cm³
- Volume of beam b = 5500 cm³
- Volume of beam c = 7500 cm³
Explanation
To find the volume of each steel beam, we use the formula:
Volume = Cross-sectional Area × Length
The problem provides the cross-section dimensions in centimeters (cm) and the length of the beams in meters. To calculate the volume in cubic centimeters (cm³), we must first convert the length to centimeters.
Length = 5 meters = 5 × 100 = 500 cm.
Next, we calculate the cross-sectional area for each of the three beams.
Beam a (L-shape):
We can split this shape into two rectangles to find its area.
- A vertical rectangle with dimensions 1 cm wide by (5 cm + 1 cm) = 6 cm tall. Its area is 1 × 6 = 6 cm².
- A horizontal rectangle with dimensions 4 cm long by 1 cm tall. Its area is 4 × 1 = 4 cm².
The total cross-sectional area is 6 cm² + 4 cm² = 10 cm².
Volume a = 10 cm² × 500 cm = 5000 cm³.
Beam b (T-shape):
This shape can also be split into two rectangles.
- The top horizontal rectangle is 6 cm wide by 1 cm tall. Its area is 6 × 1 = 6 cm².
- The vertical stem is 5 cm tall by 1 cm wide. Its area is 5 × 1 = 5 cm².
The total cross-sectional area is 6 cm² + 5 cm² = 11 cm².
Volume b = 11 cm² × 500 cm = 5500 cm³.
Beam c (U-shape):
The area can be calculated by viewing it as a large outer rectangle with a smaller inner rectangle removed.
- The area of the outer rectangle is 6 cm × 4.5 cm = 27 cm².
- The area of the inner, empty rectangle is 4 cm × 3 cm = 12 cm².
The cross-sectional area is the difference: 27 cm² – 12 cm² = 15 cm².
Volume c = 15 cm² × 500 cm = 7500 cm³.
