Problem 74 - Algebra 2

Question

Suppose you turn the water on in an empty bathtub with vertical sides. After 20 s, the water has reached a level of 1.15 in. You then leave the room. You want to turn the water off when the level in the bathtub is 8.5 in. How many minutes later should you return? (Hint: Begin by identifying two terms of an arithmetic sequence.)

Step-by-Step Solution

Answer

It will take approximately 148 seconds (or 2 minutes and 28 seconds) to fill the bathtub to 8.5 inches.

Step 1: Calculate the Rate of Change

Calculate the rate at which the water is filling the bathtub: Rate = change in water level / change in time = (1.15 inches - 0 inches) / (20 seconds - 0 seconds) = 1.15 inches / 20 seconds = 0.0575 inches/second.

Step 2: Convert Rate to Minutes

To make the calculation easier, convert the rate from inches/second to inches/minute. There are 60 seconds in a minute, so multiply 0.0575 inches/second by 60 seconds/minute to get 3.45 inches/minute.

Step 3: Calculate Time to Reach Desired Water Level

Now that you have the rate of change in a convenient unit, use it to find how long it will take the bathtub to fill up to 8.5 inches. Time = Desired level / Rate = 8.5 inches / 3.45 inches/minute = 2.464 minutes.

Step 4: Convert Time to Seconds

The final answer should be rounded to the nearest second. There are 60 seconds in a minute, so multiply 2.464 minutes by 60 seconds/minute to get about 148 seconds.

Key Concepts

Rate of Change

The rate of change is a mathematical concept that describes how one quantity changes in relation to another. In this case, it tells us how quickly the water level in the bathtub is rising over time. To determine this rate, we calculate the difference in water level after a specific time has passed. In the example given, the water level rose by 1.15 inches over 20 seconds. Therefore, the rate of change is calculated as:

• Change in water level: 1.15 inches
• Time elapsed: 20 seconds
• Rate of change: \( \frac{1.15\text{ inches}}{20\text{ seconds}} = 0.0575\text{ inches/second} \)

This rate is crucial because it helps us predict how long it will take for the water to reach a desired level by maintaining the same rate.

Water Level

The water level is the height, in inches in this scenario, at which the water has risen in the bathtub. Maintaining awareness of the water level is key to calculating how much more time is needed to reach your desired height. Initially, we know the water level is 0 inches when the tub starts filling. After 20 seconds, it reaches 1.15 inches. Our goal is to fill the tub to 8.5 inches. Here's the breakdown:

• Starting level: 0 inches
• Level after 20 seconds: 1.15 inches
• Target level: 8.5 inches

Understanding these measurements allows us to use the rate of change effectively and calculate how much time is required to reach this target height.

Unit Conversion

Unit conversion is an essential step in solving problems involving different units of measurement, like converting time from seconds to minutes. In this problem, we initially find the rate of water level increase in inches per second.To make it easier to calculate the required filling time, we convert this rate to inches per minute. Knowing there are 60 seconds in a minute, we multiply:

• Initially: \(0.0575\) inches/second
• Convert rate to minutes: \(0.0575 \times 60 = 3.45\) inches/minute

Having the rate in inches per minute simplifies calculating the time needed in minutes to fill the tub to the desired level.

Time Calculation

Time calculation involves figuring out how long it will take to reach a specific water level based on the rate of change.With the rate converted to inches per minute, we can easily find the time needed to reach 8.5 inches. The formula used is the division of the desired water level by the rate of change:

• Desired level: 8.5 inches
• Rate: 3.45 inches/minute
• Time in minutes: \( \frac{8.5}{3.45} \approx 2.464\) minutes

To ensure a precise answer, we convert the time back to seconds by noting that 1 minute equals 60 seconds. Therefore, we calculate:

• Time in seconds: \(2.464 \times 60 \approx 148\text{ seconds} \)

Rounding results appropriately helps in providing an easy-to-understand final answer.