Question

Constance is controlling the flow of a solution into a graduated cylinder. She wants to go to lunch and decides to set the flow at a lower rate rather than shut it off completely. The cylinder already holds 72 milliliters of solution and can hold a maximum of 500 milliliters. If \(r\) represents the rate at which the flow is set in milliliters per minute, which inequality could Constance solve to determine a safe range of flow rates, assuming she plans to take 60 minutes for lunch? A. \(72 r+60 \geq 500\) B. \(72 r+60 \leq 500\) C. \(60 r+72 \geq 500\) D. \(60 r+72 \leq 500\)

Short answer

The correct inequality is D: \(60r + 72 \leq 500\).

Step-by-step solution

Understand the Problem

Identify the known quantities and the unknown variables in the problem. Constance's cylinder currently holds 72 milliliters of solution and can hold up to 500 milliliters. She plans to set the flow rate, represented by variable r, for 60 minutes.

Set Up the Equation

Set up an equation representing the maximum capacity of the cylinder after 60 minutes. Constance will add solution at a rate of r milliliters per minute for 60 minutes. The cylinder already has 72 milliliters.

Write the Total Volume After 60 Minutes

The total volume of solution after 60 minutes can be written as the initial volume plus the volume added in 60 minutes: \[ 72 + 60r \]

Compare to Maximum Cylinder Capacity

Set up an inequality to ensure the total volume after 60 minutes does not exceed the cylinder's capacity of 500 milliliters: \[ 72 + 60r \leq 500 \]

Identify the Correct Inequality

By comparing the given choices with our inequality, we can see the correct inequality matches choice D: \[ 60r + 72 \leq 500 \]

Key concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values where they are not equal. Inequalities use symbols like <, >, ≤, and ≥ to express these relationships. In our exercise, we use an inequality to determine the maximum flow rate Constance should set so the cylinder does not overflow. This inequality is written as: \[ 60r + 72 \leq 500 \]
Here, \( r \) represents the flow rate in milliliters per minute, and we ensure that the total volume of the solution after 60 minutes does not surpass 500 milliliters, the cylinder's capacity. Inequalities are essential for expressing and solving problems that involve limits and ranges of values.
To solve the inequality, isolate the variable \( r \) by following these steps:

• Subtract 72 from both sides: \[ 60r \leq 428 \]
• Divide both sides by 60: \[ r \leq \frac{428}{60} \]
• Simplify the fraction if necessary to get the range of acceptable values for \( r \)

Flow Rate

Flow rate is the volume of fluid that passes a point in a given time. It's commonly represented in units such as milliliters per minute (\text{mL/min}). In the context of this exercise, Constance is adjusting the flow rate of a solution entering a cylinder.
The flow rate, denoted as \( r \), helps determine how long it will take to fill a container to a specific volume. Understanding the flow rate is crucial in scenarios involving timed filling or emptying of containers, as in this problem where the solution flows at a constant rate for a set period.
To comprehend how flow rate impacts the total volume over time, you use the formula:

• Total Volume Added = Flow Rate × Time

For Constance, the total volume added in 60 minutes at a flow rate \( r \) is given by \( 60r \). This added volume, when combined with the existing 72 milliliters, gives us the new total volume in the cylinder after an hour.

Volume Calculation

Volume calculation is a fundamental concept in mathematics and science that measures the amount of space an object or substance occupies. In the exercise, we calculate the total volume of solution in the cylinder after a certain period.
Initially, the cylinder holds 72 milliliters. Over 60 minutes, additional volumes are calculated based on the flow rate \( r \):

• Total Added Volume = Flow Rate × Time
• \( 60r \) (since time = 60 minutes)

The total volume in the cylinder after 60 minutes is a sum of the initial volume and the added volume: \[ 72 + 60r \]
This calculation ensures that the combined volume does not exceed the cylinder's maximum capacity of 500 milliliters. Volume calculation is essential in various real-world applications including fluid dynamics, cooking measurements, and industrial processes.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication). They represent general quantities without specifying their exact values. In our problem, expressions help model the flow and volume relationship.
The core algebraic expression in this exercise is: \[ 60r + 72 \]
It comprises:

• An initial volume (72 milliliters)
• An added volume (60r, where 60 represents the time in minutes and \( r \) is the flow rate per minute)

This expression quantifies the total volume Constance must manage. When solving for inequalities, algebraic expressions allow us to manipulate and isolate variables to find acceptable ranges and solutions.
Understanding and constructing algebraic expressions is crucial for solving complex problems in mathematics, science, and engineering, as they create a bridge between real-world scenarios and mathematical models.