Question

The scatterplot above shows the weight, in ounces, of the fruits on a certain truffula tree from days 55 to 85 after flowering. According to the line of best fit in the scatterplot above, which of the following is the closest approximation of the number of days after flowering of a truffula fruit that weighs \(5.75\) ounces? A) 63 B) 65 C) 77 D) 81

Short answer

To find the approximate number of days after flowering for a truffula fruit that weighs 5.75 ounces, we use the line of best fit equation \(y = mx + b\), where y is the weight in ounces, x is the number of days, m is the slope, and b is the y-intercept. Substitute y with 5.75, and rearrange the equation to solve for x: \(x = \frac{5.75-b}{m}\). Plug in the slope and y-intercept values from the scatterplot, and solve for x. Compare the calculated value to the given choices (A: 63, B: 65, C: 77, and D: 81) and choose the closest approximation.

Step-by-step solution

Understand the problem

Given the weight of the truffula fruit as 5.75 ounces. We need to find the approximate number of days after flowering for this weight using the line of best fit.

Determine the format of the line of best fit

Since we can't see the actual scatterplot, let's assume that the line of best fit equation is given in the format: \[y = mx + b\] Where y is the weight of the truffula fruit in ounces, x is the number of days after flowering, m is the slope of the line, and b is the y-intercept.

Set the weight equal to 5.75 ounces in the equation

Replace y with the given weight (5.75 ounces) in the equation: \[5.75 = mx + b\]

Rearrange the equation to solve for x

To find x, which represents the number of days after flowering, rearrange the equation: \[x = \frac{5.75 - b}{m}\]

Substitute the slope and y-intercept values

Since you can see the line of best fit in the scatter plot provided, you can determine the slope m and y-intercept b by analyzing the graph. Now, plug these values into the equation: \[x = \frac{5.75 - \text{y-intercept}}{\text{slope}}\]

Calculate the approximate number of days after flowering

Solve the equation to get the approximate number of days after flowering (x): \[x = \text{Number of days after flowering}\] Compare the calculated value to the choices provided (A: 63, B: 65, C: 77, and D: 81) and choose the closest approximation.

Key concepts

Line of Best Fit

Understanding the line of best fit, also known as a trend line, is foundational in analyzing data from scatterplots. It is a straight line that passes as closely as possible through the center of the set of points on a graph. In other words, it minimizes the distance between the line and all the points.

It serves a crucial purpose: predicting unknown values. By extending the line, you can estimate data points that lie outside the range observed. For instance, using a given fruit's weight, you can predict its days after flowering following the line's trend. When working with the equation of the line of best fit, typically presented as
\[y = mx + b\], 'm' represents the slope—indicating the steepness or incline of the line—and 'b' symbolizes the y-intercept, or where the line crosses the y-axis. Students must learn to identify these elements visually and numerically to make accurate predictions.

Interpreting Graphs

Interpreting graphs is a key skill, not just for homework exercises but for making sense of real-world data. The ability to read a scatterplot and derive meaning involves looking at the distribution of data points and understanding the relationship between the two variables presented. For example, when examining the weight of fruits against days after flowering, one can observe if there's a trend indicating a direct or inverse relationship.

The importance of this skill is recognized on standardized tests like the SAT, where students must interpret graphical information quickly and correctly. It's a step beyond just 'reading' the graph; it's analyzing to draw conclusions. This involves identifying clusters, gaps, and outliers in the data points, as these can affect the accuracy of the line of best fit and the conclusions drawn from it.

SAT Mathematics Preparation

The SAT includes a mathematics section that tests a range of skills, including data analysis. Effective preparation for this test involves familiarizing oneself with interpreting diverse types of graphs and the mathematical concepts underlying them, such as the line of best fit. Additionally, being agile in transforming visual data into algebraic equations is essential.

To prepare, students should practice using real SAT questions to get comfortable with the format and time constraints. They should also undertake exercises like the provided example, where they use a scatterplot to compute the days to certain fruit weights. Grasping these concepts well not only boosts SAT scores but also equips students with analytical tools that are valuable in academic and professional realms alike.

Data Analysis

Data analysis is an encompassing term that involves examining, cleaning, transforming, and modeling data to discover useful information and support decision-making. In our context, creating and interpreting a scatterplot are initial steps in the data analysis process. The line of best fit plays a pivotal role as it helps model the relationship between variables.

When students determine the line equation, they engage in a type of simple linear regression. This concept isn't limited to academics; it's used widely in fields such as economics, biology, and engineering to predict trends and outcomes. Data analysis skills enable students to not only excel in their classes but also to interpret the vast amounts of data they encounter in their daily lives.