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Chapter 11: Problem 2

A cylindrical storage tank has a radius of 1.22 m. When filled to a height of 3.71 m, it holds 14 300 kg of a liquid industrial solvent. What is the density of the solvent?

Short Answer

Expert verified
The density of the solvent is approximately 827.67 kg/m鲁.

Step by step solution

01

Understand the Problem

The problem involves a cylindrical tank filled with a liquid, and we need to find the density of the liquid. We know the radius of the tank (1.22 m), the height the liquid fills (3.71 m), and the mass of the liquid (14,300 kg).
02

Calculate the Volume of the Cylinder

The volume of a cylinder is calculated using the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Here, \( r = 1.22 \) m and \( h = 3.71 \) m. Therefore, \[ V = \pi (1.22)^2 (3.71) \].
03

Perform the Volume Calculation

Using the values \( r = 1.22 \) m and \( h = 3.71 \) m, the volume is calculated as:\[ V = \pi (1.22)^2 (3.71) \approx 17.27 \text{ cubic meters} \].
04

Density Formula

The density \( \rho \) is the mass divided by the volume of the substance. The formula is \( \rho = \frac{m}{V} \), where \( m \) is the mass and \( V \) is the volume. We know \( m = 14300 \) kg and from Step 3, \( V \approx 17.27 \) m鲁.
05

Calculate the Density

Substitute the known values into the density formula: \[ \rho = \frac{14300}{17.27} \approx 827.67 \text{ kg/m}^3 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylindrical Tank
A cylindrical tank is a container with a circular base and straight sides, much like a giant can. These tanks are often used to store liquids because their shape allows for efficient use of space and structural integrity.

Key characteristics of a cylindrical tank include:
  • The radius, which is the distance from the center of the circular base to the outer edge.
  • The height, which is the distance from the base to the top of the cylinder.
  • The volume, which is the amount of space inside the tank.
In our exercise, the cylindrical tank had a radius of 1.22 meters and a height filled to 3.71 meters. These dimensions are critical for calculating the volume of the tank, which leads us to our next key concept.
Volume Calculation
Calculating the volume of a cylinder is straightforward if you remember the formula: \[ V = \pi r^2 h \] Where:
  • \( V \) = Volume of the cylinder
  • \( r \) = Radius of the cylinder鈥檚 base
  • \( h \) = Height of the cylinder
  • \( \pi \) = Math constant approximately equal to 3.14159
For our cylindrical tank with a radius of 1.22 meters and a height of 3.71 meters, the volume calculation involves plugging these values into our formula: \[ V = \pi (1.22)^2 (3.71) \] After carrying out the multiplication and considering \( \pi \), we find that the tank's volume is approximately 17.27 cubic meters. Understanding this volume is essential as it helps us proceed to calculating the density of the liquid, which relates the mass to this calculated volume.
Mass and Volume Relationship
The relationship between mass and volume is fundamentally explained by a substance's density. Density provides a way to compare how much matter is in a given space. Mathematically, density \( \rho \) is expressed as: \[ \rho = \frac{m}{V} \] Where:
  • \( \rho \) = Density
  • \( m \) = Mass
  • \( V \) = Volume
In our scenario, the industrial solvent filling the cylindrical tank has a mass of 14,300 kg. We calculated its volume to be approximately 17.27 m鲁. Substituting these values into the density formula gives: \[ \rho = \frac{14300}{17.27} \approx 827.67 \text{ kg/m}^3 \] Knowing the density of a substance is useful in many real-world applications, such as designing storage tanks, ensuring structural integrity, and fulfilling industrial requirements.

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Most popular questions from this chapter

The drawing shows a hydraulic system used with disc brakes. The force \(\overrightarrow{\mathbf{F}}\) is applied perpendicularly to the brake pedal. The pedal rotates about the axis shown in the drawing and causes a force to be applied perpendicularly to the input piston (radius \(=9.50 \times 10^{-3} \mathrm{m} )\) in the master cylinder. The resulting pressure is transmitted by the brake fluid to the output plungers (radii \(=1.90 \times 10^{-2} \mathrm{m}\) ), which are covered with the brake linings. The linings are pressed against both sides of a disc attached to the rotating wheel. Suppose that the magnitude of \(\overrightarrow{\mathbf{F}}\) is 9.00 \(\mathrm{N}\) . Assume that the input piston and the output plungers are at the same vertical level, and find the force applied to each side of the rotating disc.

A mercury barometer reads 747.0 mm on the roof of a building and 760.0 \(\mathrm{mm}\) on the ground. Assuming a constant value of 1.29 \(\mathrm{kg} / \mathrm{m}^{3}\) for the density of air, determine the height of the building.

A suitcase (mass \(m=16 \mathrm{kg} )\) is resting on the floor of an elevator. The part of the suitcase in contact with the floor measures 0.50 \(\mathrm{m} \times 0.15 \mathrm{m}\) . The elevator is moving upward with an acceleration of magnitude 1.5 \(\mathrm{m} / \mathrm{s}^{2}\) . What pressure (in excess of atmospheric pressure) is applied to the floor beneath the suitcase?

A hydrometer is a device used to measure the density of a liquid. It is a cylindrical tube weighted at one end, so that it floats with the heavier end downward. The tube is contained inside a large 鈥渕edicine dropper,鈥� into which the liquid is drawn using the squeeze bulb (see the drawing). For use with your car, marks are put on the tube so that the level at which it floats indicates whether the liquid is battery acid (more dense) or antifreeze (less dense). The hydrometer has a weight of \(W=5.88 \times 10^{-2} \mathrm{N}\) and a cross-sectional area of tw \(A=7.85 \times 10^{-5} \mathrm{m}^{2} .\) How far from the bottom of the tube should the mark be put that denotes (a) battery acid \(\left(\rho=1280 \mathrm{kg} / \mathrm{m}^{3}\right)\) and (b) antifreeze \(\left(\rho=1073 \mathrm{kg} / \mathrm{m}^{3}\right) ?\)

A spring is attached to the bottom of an empty swimming pool, with the axis of the spring oriented vertically. An 8.00-kg block of wood \(\left(\rho=840 \mathrm{kg} / \mathrm{m}^{3}\right)\) is fixed to the top of the spring and compresses it. Then the pool is filled with water, completely covering the block. The spring is now observed to be stretched twice as much as it had been compressed. Determine the percentage of the block's total volume that is hollow. Ignore any air in the hollow space.
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