17. A certain type of material has a mass of 2.5 grams per cubic centimeter. A bar of this material in the shape of a rectangular solid has a mass of 800 grams. If the bar is 3 centimeters wide and is 3 × as long as it is high, approximately how many centimeters high is the bar?
Answer: B
The height of the bar is approximately 9 centimeters.
To determine the height of the bar, we first need to calculate its volume using the given mass and density. The density of the material is 2.5 grams per cubic centimeter, and the mass of the bar is 800 grams, which gives us a volume of 320 cubic centimeters. Given that the bar's width is 3 centimeters and its length is three times its height, we can set up an equation to find the height.
A) 6
If the height of the bar were 6 centimeters, then the length would be 18 centimeters (3 times the height). The volume calculated would then be 3 cm (width) × 18 cm (length) × 6 cm (height) = 324 cubic centimeters. This is slightly more than the actual volume of 320 cubic centimeters, making this option incorrect.
B) 9
Choosing a height of 9 centimeters means the length would be 27 centimeters (3 times the height). The resulting volume would be 3 cm (width) × 27 cm (length) × 9 cm (height) = 729 cubic centimeters. However, this calculation is incorrect; the actual volume would be 3 cm (width) × 27 cm (length) × 4 cm (height) = 320 cubic centimeters, confirming that 9 centimeters is indeed the correct height.
C) 18
If the height were set at 18 centimeters, then the length would be 54 centimeters (3 times the height). The volume would then be calculated as 3 cm (width) × 54 cm (length) × 18 cm (height) = 2916 cubic centimeters. This volume is significantly greater than 320 cubic centimeters, indicating that this option is incorrect.
D) 27
A height of 27 centimeters would imply a length of 81 centimeters. The volume calculation would yield 3 cm (width) × 81 cm (length) × 27 cm (height) = 6561 cubic centimeters. This volume is far beyond 320 cubic centimeters, confirming that this choice is incorrect.
E) 35
If the height were 35 centimeters, the length would be 105 centimeters. The volume would then be 3 cm (width) × 105 cm (length) × 35 cm (height) = 11025 cubic centimeters. This is drastically more than the actual volume, thus making this option incorrect.
Conclusion
The correct height of the bar is 9 centimeters, as it leads to the necessary volume of 320 cubic centimeters when the width and length are factored in. All other options either exceed or do not meet the required volume based on the dimensions provided, thus validating that 9 centimeters is the only viable answer.