Question

A cape is designed for Lady Gaga's concert with 4560 sequins. If a sequin has a volume of \(0.0241 \mathrm{~cm}^{3}\) and the sequins have a density of \(41.6 \mathrm{~g} / \mathrm{cm}^{3}\), what is the mass in \(\mathrm{lb}\) and \(\mathrm{kg}\) of sequins on Lady Gaga's cape?

Short answer

The mass of sequins on the cape is approximately 4.57 kg or 10.08 lb.

Step-by-step solution

- Find the total volume of all sequins

First, calculate the total volume of all 4560 sequins by multiplying the number of sequins by the volume of one sequin: \[ \text{Total Volume} = 4560 \times 0.0241 \text{ cm}^3 \ \text{Total Volume} = 109.896 \text{ cm}^3 \]

- Calculate the total mass in grams

Next, find the total mass in grams by using the density of the sequins. Multiply the total volume by the density: \[ \text{Total Mass (g)} = 109.896 \text{ cm}^3 \times 41.6 \text{ g/cm}^3 \ \text{Total Mass (g)} = 4571.6736 \text{ g} \]

- Convert grams to kilograms

Now convert the mass from grams to kilograms by dividing by 1000: \[ \text{Total Mass (kg)} = \frac{4571.6736}{1000} \ \text{Total Mass (kg)} = 4.5716736 \text{ kg} \]

- Convert kilograms to pounds

Finally, convert the mass from kilograms to pounds using the conversion factor where 1 kg = 2.20462 lbs: \[ \text{Total Mass (lb)} = 4.5716736 \text{ kg} \times 2.20462 \ \text{Total Mass (lb)} = 10.0806 \text{ lb} \]

Key concepts

Volume

Volume is the amount of space that an object occupies. In the context of the exercise, volume is associated with the size of a single sequin. Here, we find that each sequin occupies a volume of 0.0241 cm³. To determine the total volume of all sequins, multiply the volume of one sequin by the number of sequins:
\[ \text{Total Volume} = 4560 \times 0.0241 \text{ cm}^{3} = 109.896 \text{ cm}^{3} \]
So, all 4560 sequins together take up 109.896 cm³ of space. Understanding volume is crucial because it helps in further calculations, such as determining the mass when combined with density.

Density

Density can be thought of as how heavy something is for its size. It’s calculated as mass per unit volume. The formula for density is:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
In this exercise, the density of each sequin is given as 41.6 g/cm³. Once we know the total volume of the sequins, we can find the total mass using this density. By multiplying the total volume by the density:
\[ \text{Total Mass (g)} = 109.896 \text{ cm}^3 \times 41.6 \text{ g/cm}^3 = 4571.6736 \text{ g} \]
Hence, 109.896 cm³ of sequins has a total mass of 4571.6736 grams. Knowing density helps determine how much material there is in a given volume.

Mass Conversion

Mass conversion involves changing the mass of an object from one unit to another. In the given exercise, after finding the mass in grams, students need to convert it into kilograms and pounds.

1. **Grams to Kilograms:** Since 1 kilogram equals 1000 grams, divide the mass in grams by 1000.\[ \text{Total Mass (kg)} = \frac{4571.6736}{1000} = 4.5716736 \text{ kg} \] 2. **Kilograms to Pounds:** To convert kilograms to pounds, use the conversion factor where 1 kilogram is equal to approximately 2.20462 pounds. Multiply the mass in kilograms by 2.20462.
\[ \text{Total Mass (lb)} = 4.5716736 \text{ kg} \times 2.20462 = 10.0806 \text{ lb} \] Understanding mass conversion is essential for working in different measurement systems.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another. This is a crucial skill in science and everyday life. Let's break it down with some examples from the exercise.

1. **Volume Conversion (if needed):** Though the exercise sticks to cm³, you might come across situations where you need to convert volumes, like from cm³ to m³. For example, 1 m³ = 1,000,000 cm³.

2. **Mass Conversion:** In the exercise, we converted mass from grams to kilograms and then to pounds. Remember the key factors:

• 1 kg = 1000 g
• 1 kg = 2.20462 lbs

Steps for converting: \[ \text{Mass (kg)} = \frac{\text{Mass (g)}}{1000} \] \[ \text{Mass (lb)} = \text{Mass (kg)} \times 2.20462 \]
When you understand unit conversion, you can easily switch between different measurement units, making it simpler to interpret and communicate scientific data.