Question

The graph above represents the reaction rate, \(r\), at which an unfinished iron nail rusts in water during the first 10 days of an experiment, where \(d\) gives time measured in days. What was the total amount of rust produced from \(d=2\) to \(d=6\) ? A) \(0.8\) grams B) \(1.6\) grams C) \(2.4\) grams D) \(3.2\) grams

Short answer

The short answer for this question is: To find the total amount of rust produced between day 2 and day 6, first read the graph and determine the reaction rate ($r_d$) for each day within the range. Then, calculate the area under the curve by summing the rectangle areas: \[Area_{total} = \sum_{d=2}^{6} r_d\] Finally, compare your calculated rust amount with the provided choices (A, B, C, or D) to find the correct answer.

Step-by-step solution

Read the graph and determine the reaction rate for day 2 to day 6

First, carefully read the graph, noting both the day and reaction rate. The graph should be plotted with days on the x-axis and reaction rate (r) on the y-axis. Using the graph, record the reaction rate for each day between day 2 and day 6. #step 2:

Calculate the area under the curve

To find the total amount of rust produced during day 2 through day 6, sum the areas of the rectangles under the curve. For this, use the values of r and a width of 1 day to calculate the rectangle areas: \[Area_{total} = \sum_{d=2}^{6} r_d\] #step 3:

Compare the calculated rust amount with provided choices

Once you have found the total amount of rust produced by calculating the area, compare your results with the given choices in the problem statement to determine which of the alternative (A, B, C, or D) is the correct answer.

Key concepts

Interpreting Graphs

Visual representations like graphs are essential tools for communicating complex data in a digestible manner. When it comes to interpreting graphs, the key is to understand what each axis represents. In SAT math problems, you might encounter graphs that depict various functions, including linear, quadratic, or in this case, the rate of a chemical reaction. A correctly interpreted graph provides a visual story of the relationship between variables.

For instance, in the context of our problem, the x-axis shows the time in days, while the y-axis indicates the reaction rate of iron rusting. By reading the graph, you can deduce how quickly the nail rusts at different time intervals. This skill is not only crucial for SAT success but also for making informed decisions based on data in real-world scenarios.

Reaction Rate

The reaction rate is a measure of how quickly a chemical reaction occurs and is usually expressed in terms of the amount of reactant consumed or the amount of product formed over time. In the context of our SAT problem, the reaction rate indicates how fast the iron nail rusts, which means how many grams of rust are formed per day.

Understanding the concept of reaction rate is vital in chemistry, and it also pops up in various real-life situations. For instance, in the pharmaceutical industry, reaction rates can determine how quickly a medication takes effect. On the SAT, a clear comprehension of reaction rates and how they're represented on graphs can significantly enhance your ability to solve related problems accurately.

Area Under Curve

The area under the curve in a graph can represent several different concepts, such as distance traveled over time, or, as in our SAT problem, the total amount of substance produced or consumed in a reaction. To find the area under a curve on a graph, you typically sum the areas of shapes that you can clearly define and measure. In scenarios where the graph displays a rate, like a reaction rate, this area signifies an accumulated quantity, such as the total amount of product formed over a specific period.

For our SAT Math problem, by calculating the area of rectangles under the reaction rate curve from day 2 to day 6, we determine the total rust produced by an iron nail. This method of using the area under the curve is a fundamental concept in calculus known as integration, but for the SAT, you'll often use simpler geometric methods to estimate this area.

SAT Practice Questions

Preparing for the SAT involves practicing with a variety of questions that assess your mathematical understanding and your ability to apply these concepts to solve problems. SAT practice questions typically cover a comprehensive range of topics, from algebra to data analysis, like the one in our example. To excel at SAT math, you should regularly practice interpreting graphs, understanding rates, and calculating areas under curves.

Each practice question on the SAT is designed to test specific skills. In our rusting iron nail problem, the skills are interpreting a graph and applying knowledge of areas under the curve. Keep in mind that practice questions not only reinforce mathematical principles but also help you become more familiar with the test's format, style, and time constraints. This makes tackling real SAT questions on test day a less daunting task.