Question

Vegetable oil for cooking is dispensed from a cylindrical can fitted with a spray nozzle. According to the label, the can is able to deliver 560 sprays, each of duration \(0.25 \mathrm{~s}\) and each having a mass of \(0.25 \mathrm{~g}\). Determine (a) the mass flow rate of each spray, in \(\mathrm{g} / \mathrm{s}\). (b) the mass remaining in the can after 560 sprays, in \(\mathrm{g}\), if the initial mass in the can is \(170 \mathrm{~g}\).

Short answer

(a) 1 g/s (b) 30 g

Step-by-step solution

Identify the mass of each spray

The mass of each spray is given as 0.25 g according to the problem statement.

Identify the duration of each spray

The duration of each spray is given as 0.25 s according to the problem statement.

Calculate the mass flow rate

Use the formula for the mass flow rate: \(\text{Mass Flow Rate} = \frac{\text{Mass of each spray}}{\text{Duration of each spray}}\). Plugging in the values: \(\text{Mass Flow Rate} = \frac{0.25 \, \text{g}}{0.25 \, \text{s}} = 1 \, \text{g/s}\).

Calculate the total mass dispensed in 560 sprays

Multiply the mass of each spray by the number of sprays: \(\text{Total Mass Dispensed} = 0.25 \, \text{g} \times 560 = 140 \, \text{g}\).

Determine the initial mass in the can

The initial mass in the can is given as 170 g according to the problem statement.

Calculate the mass remaining in the can after 560 sprays

Subtract the total mass dispensed from the initial mass in the can: \(\text{Mass Remaining} = 170 \, \text{g} - 140 \, \text{g} = 30 \, \text{g}\).

Key concepts

mass flow rate

The mass flow rate is an essential concept in fluid dynamics and various engineering applications. It measures the mass of a substance passing through a cross-sectional area per unit time. In this exercise, we calculate the mass flow rate of vegetable oil dispensed through a spray nozzle.
Given, the mass of each spray is 0.25 grams, and the duration of each spray is 0.25 seconds. To find the mass flow rate, we use the formula:
\(\text{Mass Flow Rate} = \frac{\text{Mass of each spray}}{\text{Duration of each spray}}\)
By plugging in the values:
\(\text{Mass Flow Rate} = \frac{0.25 \, \text{g}}{0.25 \, \text{s}} = 1 \, \text{g/s}\)
This tells us that 1 gram of oil is sprayed every second.

mass conservation

Mass conservation is a fundamental principle in physics and engineering. It states that mass cannot be created or destroyed in a closed system, only transferred. This means the total mass before and after an event remains constant.
In our problem, we apply mass conservation to determine the mass remaining in the can after a number of sprays. Initially, the can contains 170 grams of oil. Each spray dispenses 0.25 grams of oil, and there are 560 sprays:
\(\text{Total Mass Dispensed} = 0.25 \, \text{g} \times 560 = 140 \, \text{g}\)
By deducting the total mass dispensed from the initial mass, we find the mass remaining:
\(\text{Mass Remaining} = 170 \, \text{g} - 140 \, \text{g} = 30 \, \text{g}\)
So, after 560 sprays, 30 grams of oil remain in the can.

unit conversion

Unit conversion is crucial in solving problems where different units of measurement are involved. While this exercise uses consistent units (grams and seconds), other scenarios may require conversions. For instance, switching between mass units like grams, kilograms, and pounds or time units like seconds, minutes, and hours.
To perform a unit conversion, you multiply the quantity by a conversion factor. For example, converting grams to kilograms:
\(\text{Weight in kg} = \text{Weight in grams} \times 0.001\)
Or, converting seconds to minutes:
\(\text{Time in minutes} = \text{Time in seconds} \times \frac{1}{60}\)
Ensuring all quantities use appropriate units simplifies calculations and reduces errors.

problem-solving steps

Effective problem-solving involves breaking down a problem into manageable steps, as demonstrated in this exercise. Here are the general steps followed:
1. **Understand the Problem**: Identify the given data and the questions asked.
2. **Identify Key Information**: Note down the known values. Here, the mass per spray (0.25 g) and duration of each spray (0.25 s).
3. **Use Formulas**: Apply relevant formulas. For mass flow rate, use:
\(\text{Mass Flow Rate} = \frac{\text{Mass of each spray}}{\text{Duration of each spray}}\)
4. **Perform Calculations**: Calculate the values step by step. First, determine the mass flow rate (1 g/s), then the total mass dispensed (140 g), and finally the remaining mass (30 g).
5. **Verify**: Double-check calculations and logic to ensure accuracy.
Using structured steps like these helps solve complex problems systematically and accurately.