Question

The radius (r) of the international reference kilogram cylinder is \(1.95 \mathrm{~cm} .\) Assuming the density of the kilogram is \(21.50 \mathrm{~g} / \mathrm{cm}^{3},\) calculate its height \((\mathrm{h}) .\) The volume of a cylinder equals \(\pi \mathrm{r}^{2} \mathrm{~h},\) where \(\pi\) is the constant 3.14.

Short answer

The height of the cylinder is approximately 3.90 cm.

Step-by-step solution

Understand the Problem

We need to calculate the height of a cylinder given its radius and density. The equation for cylinder volume is \( V = \pi r^2 h \). We are given the radius of the cylinder and its density, and we know the mass is 1000 g (1 kg).

Calculate the Volume of the Cylinder

Since the mass \( m \) is 1000 grams and the density \( \rho \) is \( 21.50 \ \text{g/cm}^3 \), we calculate the volume \( V \) using the formula \( V = \frac{m}{\rho} \). Thus, the volume is \[ V = \frac{1000}{21.50} \approx 46.51 \ \text{cm}^3. \]

Use the Volume Formula to Find the Height

Using the volume formula \( V = \pi r^2 h \), we can solve for \( h \). Substitute \( V = 46.51 \ \text{cm}^3 \), \( r = 1.95 \ \text{cm} \), and \( \pi = 3.14 \). The formula becomes \[ 46.51 = 3.14 \times (1.95)^2 \times h. \]

Calculate the Height

First, calculate \( r^2 \) which is \( (1.95)^2 = 3.8025 \). Now substitute in the equation: \[ 46.51 = 3.14 \times 3.8025 \times h. \] Divide both sides by \( 3.14 \times 3.8025 \) to solve for \( h \): \[ h = \frac{46.51}{3.14 \times 3.8025} \approx 3.90 \ \text{cm}. \]

Key concepts

Cylinder Volume

Understanding how to calculate the volume of a cylinder is essential in various fields such as physics, engineering, and chemistry. A cylinder is a 3-dimensional shape with two parallel circular bases and a curved surface connecting them. The formula to calculate the volume of a cylinder is given by:\[ V = \pi r^2 h \]where:

• \( V \) is the volume of the cylinder,
• \( \pi \) is a constant approximately equal to 3.14,
• \( r \) is the radius of the circular base,
• \( h \) is the height of the cylinder.

To find the volume, you need to know the radius of the base and the height of the cylinder. First, square the radius, multiply it by \( \pi \), and then multiply by the height. This is a straightforward application of multiplication and squaring, providing a simple yet powerful tool for calculating the space inside cylindrical objects.

Density and Mass

Density is a fundamental property of matter that describes how much mass is contained in a given volume. It is often represented by the formula:\[ \rho = \frac{m}{V} \]where:

• \( \rho \) is the density,
• \( m \) is the mass,
• \( V \) is the volume.

The units of density are typically grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). In practical terms, if you know the mass and density of a substance, you can find its volume using the rearranged formula:\[ V = \frac{m}{\rho} \]This relationship helps in determining not only the volume but also other characteristics of a material or object. It's essential when working with items made from metals or assessing purity, as well as in a variety of scientific calculations in chemistry and physics.

Basic Measurements in Chemistry

Chemistry relies heavily on accurate measurements to analyze chemical substances and reactions effectively. Understanding fundamental units like mass, volume, and density is crucial to solving chemical problems and performing experiments. Some of the key measurements in chemistry include:

• Mass: Indicates how much matter an object contains. It is usually measured in grams or kilograms.
• Volume: Refers to the space an object occupies, measurable in liters or cubic centimeters.
• Density: Represents mass per unit volume, offering insights into a substance's compactness.

In practical terms, these measurements allow chemists to calculate quantities in reactions, determine concentration levels, and assess the purity of compounds. Enhancing problem-solving skills in making these measurements aids in accurately evaluating chemical properties and reactions.