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Chapter 1: Problem 19

The scatterplot above shows the height in centimeters for both the drop and bounce of eight different balls of the same type. The line of best fit for the data is also shown. According to the line of best fit, which of the following is closest to the predicted increase in bounce height, in centimeters, for every increase of 100 centimeters in drop height? A) 25 B) 20 C) 15 D) 10

Short Answer

Expert verified
Using the line of best fit in the scatterplot, calculate the slope between two points on the line: \(Slope = \frac{change \, in \, bounce \, height}{change \, in \, drop \, height}\). Multiply the slope by 100 cm to find the predicted increase in bounce height for every 100 cm increase in drop height: \(Predicted \, increase \, in \, bounce \, height = Slope \times 100 \, cm\). Compare the result with the given options and choose the closest option.

Step by step solution

01

Identify the scale on both axes

Begin by identifying the scale used on both the horizontal (drop height) and vertical (bounce height) axes of the scatterplot. We need to determine what increments were used for both drop and bounce height, as this will help determine the slope of the line of best fit.
02

Determine the slope of the line of best fit

The slope of the line of best fit is the vertical change (increase in bounce height) divided by the horizontal change (increase in drop height) between two points on the line. To calculate the slope, find two points on the line of best fit and use the formula: \[Slope = \frac{change \, in \, bounce \, height}{change \, in \, drop \, height}\] Select two points on the line of best fit and calculate the slope using the formula.
03

Calculate the increase in bounce height for an increase of 100 centimeters in drop height

Once you have the calculated the slope, we can use it to determine the increase in bounce height for every 100 centimeters increase in drop height. \[Predicted \, increase \, in \, bounce \, height = Slope \times 100 \, cm\] Multiply the slope by 100 cm to find the predicted increase in bounce height.
04

Compare your calculated value to the given options

Compare the calculated predicted increase in bounce height to the provided answer choices (A) 25, (B) 20, (C) 15, (D) 10. Choose the option that is closest to your calculated value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatterplot Analysis
When examining a scatterplot, the goal is to determine how two variables relate to each other. In the case of ball drop and bounce heights, each point on the plot represents an individual ball's performance. Analyzing this scatterplot involves identifying trends and outliers.

A trend shows whether there is a correlation, wherein the bounce height tends to increase with an increase in drop height. To convey this visually, a line of best fit, or a trend line, is drawn to summarize the relationship between the two variables.

Understanding the scatterplot is crucial for interpretation. It's important to verify the scales on each axis to ensure accurate calculations. When predicting future values, the trend line provides insight. In the context of SAT math preparation, being proficient in scatterplot analysis can significantly aid in solving data interpretation questions effectively.
Slope Calculation
The slope of a line on a graph reveals the rate of change between two variables. In mathematics, the slope is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points. By analyzing the line of best fit, we use the formula: \[Slope = \frac{rise}{run}\].

In this particular exercise, the rise represents the change in bounce height, while the run corresponds to the change in drop height. Selecting two clear points on the line of best fit allows us to calculate the slope. This slope then helps to estimate the increase in bounce for every 100 cm increase in drop height, a key concept for handling real-world proportional relationships in SAT math problems.
SAT Mathematics Preparation
Preparing for the SAT Mathematics section involves understanding various types of algebraic and graphical concepts, including the analysis of scatterplots and the calculation of slopes. Regular practice with exercises like the provided example assists students in honing these skills.

It's important for students to not only solve problems but also understand the concepts behind the steps. This in-depth knowledge empowers students to tackle SAT questions with confidence and achieve higher scores.
  • Grasp the scales and labeling of axes
  • Master slope calculation for lines of best fit
  • Translate graphical analysis into problem-solving scenarios
By breaking down complex questions into manageable steps and familiarizing oneself with a variety of problem types, students can improve their performance on the actual test.

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