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Chapter 1: Problem 57

The following densities are given at \(20^{\circ} \mathrm{C}\) : water, \(0.998 \mathrm{g} / \mathrm{cm}^{3} ;\) iron, \(7.86 \mathrm{g} / \mathrm{cm}^{3} ;\) aluminum, \(2.70 \mathrm{g} / \mathrm{cm}^{3}\). Arrange the following items in terms of increasing mass. (a) a rectangular bar of iron,$$81.5 \mathrm{cm} \times 2.1 \mathrm{cm} \times 1.6 \mathrm{cm}$$ (b) a sheet of aluminum foil,$$12.12 \mathrm{m} \times 3.62 \mathrm{m} \times 0.003 \mathrm{cm}$$ (c) 4.051 L of water

Short Answer

Expert verified
The items arranged in order of increasing mass are: the aluminum sheet, the iron bar, and then the water.

Step by step solution

01

Calculate the volume and mass of the iron bar

First, we need to calculate the volume of the iron bar. This can be done by multiplying its dimensions: \(81.5 \, \mathrm{cm} \times 2.1 \, \mathrm{cm} \times 1.6 \, \mathrm{cm} = 273.42\, \mathrm{cm}^{3}\). Next, we multiply the volume we just calculated by the density of iron (7.86 \, \mathrm{g} / \, \mathrm{cm}^{3}) to give the mass: \(273.42 \, \mathrm{cm}^{3} \times 7.86 \, \mathrm{g} / \mathrm{cm}^{3} = 2149.16 \, \mathrm{g}\).
02

Calculate the volume and mass of the aluminum sheet

Similarly, we find the volume of the aluminum sheet by multiplying its dimensions: \(12.12 \, \mathrm{m} \times 3.62 \, \mathrm{m} \times 0.003 \, \mathrm{cm} = 132.10 \, \mathrm{cm}^{3}\). We then multiply that by the density of aluminum (2.70 \, \mathrm{g} / \, \mathrm{cm}^{3}) to give the mass: \(132.10 \, \mathrm{cm}^{3} \times 2.70 \, \mathrm{g} / \mathrm{cm}^{3} = 356.67 \, \mathrm{g}\). Note that we converted the dimensions from meters to centimeters so that the units match the density's units.
03

Calculate the mass of the water

Here, you can use the density directly because the volume is provided in liters and 1 liter is equivalent to 1000 cm\(^3\). So, to find the mass, multiply the volume by the density: \(4.051 \, \mathrm{L} \times 0.998 \, \mathrm{g}/\mathrm{cm}^{3} = 4042.948 \, \mathrm{g}\).
04

Arrange the items in order of increasing mass

Finally, arrange the items in order of increasing mass: Aluminum Sheet (356.67 g) < Iron Bar (2149.16 g) < Water (4042.948 g).

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Most popular questions from this chapter

The fact that the volume of a fixed amount of gas at a fixed temperature is inversely proportional to the gas pressure is an example of (a) a hypothesis; (b) a theory; (c) a paradigm; (d) the absolute truth; (e) a natural law.

In normal blood, there are about \(5.4 \times 10^{9}\) red blood cells per milliliter. The volume of a red blood cell is about \(90.0 \times 10^{-12} \mathrm{cm}^{3},\) and its density is \(1.096 \mathrm{g} / \mathrm{mL}\) How many liters of whole blood would be needed to collect \(0.5 \mathrm{kg}\) of red blood cells?

A lump of pure copper weighs \(25.305 \mathrm{g}\) in air and 22.486 g when submerged in water \((d=0.9982 \mathrm{g} / \mathrm{mL})\) at \(20.0^{\circ} \mathrm{C} .\) Suppose the copper is then rolled into a \(248 \mathrm{cm}^{2}\) foil of uniform thickness. What will this thickness be, in millimeters?

Perform the following conversions from non-SI to SI units. (Use information from the inside back cover, as needed.) (a) 68.4 in. \(=\)________cm (b) \(94 \mathrm{ft}=\)________m (c) \(1.42 \mathrm{lb}=\)________g (d) \(248 \mathrm{lb}=\)________kg (e) \(1.85 \mathrm{gal}=\)________dm\(^3\) (f) \(3.72 \mathrm{qt}=\)________mL

The following equation can be used to relate the density of liquid water to Celsius temperature in the range from \(0^{\circ} \mathrm{C}\) to about \(20^{\circ} \mathrm{C}:\) $$d\left(\mathrm{g} / \mathrm{cm}^{3}\right)=\frac{0.99984+\left(1.6945 \times 10^{-2} t\right)-\left(7.987 \times 10^{-6} t^{2}\right)}{1+\left(1.6880 \times 10^{-2} t\right)}$$ (a) To four significant figures, determine the density of water at \(10^{\circ} \mathrm{C}\). (b) At what temperature does water have a density of \(0.99860 \mathrm{g} / \mathrm{cm}^{3} ?\) (c) In the following ways, show that the density passes through a maximum somewhere in the temperature range to which the equation applies. (i) by estimation (ii) by a graphical method (iii) by a method based on differential calculus
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