Problem 45 - College Algebra with Corequisite Support

Question

A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3: 40 p.m. At this time, he started pumping gas into the tank. At exactly \(3: 44,\) the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank?

Step-by-Step Solution

Answer

The average rate of flow is 2.675 gallons per minute.

Step 1: Determine Time Duration

The problem provides the start and end times for the gas filling, from 3:40 p.m. to 3:44 p.m. Calculate the duration: 3:44 p.m. - 3:40 p.m. = 4 minutes.

Step 2: Identify Total Volume of Gasoline

The total amount of gasoline pumped is given as 10.7 gallons. This is the volume of gasoline that flowed into the tank during the 4 minutes.

Step 3: Calculate Average Rate of Flow

The average rate of flow is the total volume of gasoline divided by the time duration. Use the formula: \( \text{Rate} = \frac{\text{Volume}}{\text{Time}} \). Substitute the known values: \( \text{Rate} = \frac{10.7 \text{ gallons}}{4 \text{ minutes}} = 2.675 \text{ gallons per minute} \).

Key Concepts

Time Calculation

When determining how long it took for something to happen, understanding time calculation is crucial. In this exercise, the driver started filling gas at 3:40 p.m. and finished at 3:44 p.m. To find the duration, we subtract the start time from the end time. This gives us a period of 4 minutes. Here's how you can easily calculate such durations:

• Write down the start and end time.
• Make sure both are in the same time format (e.g., p.m., 24-hour notation).
• Subtract the minutes of the start time from the minutes of the end time, and do the same for the hours, when needed.

In our case, since both times are within the same hour, it's simply 44 minus 40, resulting in 4 minutes. Time calculations like these help understand how long an event lasts, which is fundamental in many real-world situations, including calculating rates.

Volume Measurement

Volume measurement is all about determining the amount of space that a substance occupies. In the context of liquids like gasoline, it can be measured in gallons, liters, or other units. For this exercise, the driver filled up 10.7 gallons, which is equivalent to 10.7 units of liquid volume. Understanding volume measurement involves:

• Identifying the correct unit of measurement fitting the context (e.g., gallons for gasoline).
• Accurately measuring or obtaining the volume data, often from a gas pump or any calibrated device.

Accurate volume measurements ensure that you know the exact amount of a substance involved, which is important for budgeting, planning, or further calculations like determining the rate of flow.

Average Rate Formula

Calculating the average rate of something, such as the flow of gasoline into a tank, relies on understanding the average rate formula. This calculation involves dividing the total quantity of something by the time it took to change that quantity. For this exercise, the formula used is:\[\text{Rate} = \frac{\text{Volume}}{\text{Time}}\]This formula helps in determining the rate at which gasoline was pumped. With a total volume of 10.7 gallons over a time span of 4 minutes, the calculation proceeds as follows:

• Input the total volume: 10.7 gallons.
• Input the total time: 4 minutes.
• Use the formula to find the rate: \( \text{Rate} = \frac{10.7}{4} = 2.675 \text{ gallons per minute} \).

Thus, the gasoline flows at an average rate of 2.675 gallons per minute, which means 2.675 gallons of gasoline is pumped into the tank for each minute passing. The average rate formula is widely used in various contexts to simplify and quantify processes over time, aiding in efficiency evaluations and planning.