Room Type, Length, Width, Height, Area ✓ Solved

Room Room Typelength L Ftwidth W Ftheight H Ftareaa

Extracting Additional Information from Basic Measurements related to Cubic and Rectangular Type of Objects and Make Generalization on the basis of if the Length increases what would happen to listed items. Show the Steps how you calculated the following since the Columns in Table are narrow and enter ONLY final answers rounded to three decimal places in Table.

1. Area 2. Volume 3. Surface Area 4. Other ratios in last 3 columns with appropriate units.

Calculations for Cubic and Rectangular Examples

Cubic Room

1. Area = (L x W) = (10 ft x 10 ft) = 100 ft²

2. Volume = (L x W x H) = (10 ft x 10 ft x 10 ft) = 1000 ft³

3. Surface Area = 6 x (L x W) = 6 x (10 ft x 10 ft) = 600 ft²

4. Area/Volume = (100 ft² / 1000 ft³) = 0.100 ft^-1

5. Area/Surface Area = (100 ft² / 600 ft²) = 0.1667

6. Volume/Surface Area = (1000 ft³ / 600 ft²) = 1.6667 ft

Rectangular Room

1. Area = (L x W) = (25 ft x 15 ft) = 375 ft²

2. Volume = (L x W x H) = (25 ft x 15 ft x 35 ft) = 13,125 ft³

3. Surface Area = 2 x [(L x W) + (L x H) + (W x H)] = 2 x [(25 ft x 15 ft) + (25 ft x 35 ft) + (15 ft x 35 ft)] = 2 x [375 ft² + 875 ft² + 525 ft²] = 2 x [1775 ft²] = 3550 ft²

4. Area/Volume = (375 ft² / 13125 ft³) = 0.0286 ft^-1

5. Area/Surface Area = (375 ft² / 3550 ft²) = 0.1056

6. Volume/Surface Area = (13125 ft³ / 3550 ft²) = 3.704 ft

Generalizations Based on Length Increase

Cubic Room

1. Area increases with an increase in length, as area is a product of length and width.

2. Volume also increases since volume derives from the product of length, width, and height.

3. Surface area will increase, but the rate of increase depends on the dimensional changes.

4. Area/Volume decreases because as volume increases more rapidly than area when length increases substantially.

5. Area/Surface Area remains constant as surface area increases with length in a linear proportion compared to area.

6. Volume/Surface Area initially increases and can decrease if the length increases disproportionately relative to the other measurements.

Rectangular Room

1. Area increases as the length increases.

2. Volume increases similarly as it's dependent on all three dimensions.

3. Surface Area will increase; however, the relationship is not strictly linear.

4. Area/Volume decreases as length increases; this reflects that volume grows faster than the area due to the height factor.

5. Area/Surface Area generally remains constant unless the dimensions change disproportionately.

6. Volume/Surface Area follows the same pattern as previously mentioned and will exhibit varying behavior based on dimension changes.

Conclusion

This analysis shows how changes in length affect the area, volume, and surface area of cubic and rectangular shapes. Understanding these relationships allows deeper insight into geometric properties and their implications in real-world applications.

References

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