Question

According to Figure \(1,\) the 2 methods yield the same value for \(D\) when the initial speed is closest to: A. 20 \(\mathrm{mi} / \mathrm{hr}\) . B. 40 \(\mathrm{mi} / \mathrm{hr}\) C. 60 \(\mathrm{mi} / \mathrm{hr}\) D. 80 \(\mathrm{mi} / \mathrm{hr}\)

Short answer

Answer: To determine the initial speed at which the two methods yield approximately the same value for the stopping distance, follow the steps outlined in the solution. Analyze the provided graph, identify the intersection point of the two methods, and compare the speeds given in the options (20, 40, 60, and 80 mi/hr). The answer will be the speed closest to the intersection point on the graph. Please refer to the figure for the result.

Step-by-step solution

Obtain the Figure

To solve this exercise, we must obtain and analyze the figure. Since the figure is not provided here, we will assume that it is available and analyze accordingly. Make sure to have the figure at hand before proceeding with the solution.

Analyze the intersection point

Looking at the figure, we need to see which horizontal point intersects the two method lines. If the graph is well labeled we should be able to find this point but if the graph isn't clear, we may need to approximate.

Compare speeds

Now that the intersection point is identified, we need to determine which speed line (horizontal value) is closest to that point. The initial speeds given in the options are 20, 40, 60, and 80 mi/hr. Examine where the point is located horizontally along the speed axis and choose the closest speed from the given options.

Draw conclusion

After completing steps 1 to 3, we should identify the most appropriate speed for the intersection. The answer will be one of the given options (A, B, C, or D). Note that this solution assumes that the figure is available, if the figure is not available, another approach should be taken.

Key concepts

Analyzing Graphs

When tackling an ACT Math problem involving graphs, the key is to understand how to interpret the data presented visually. Graphs can seem daunting at first sight, but they are just a method of displaying information in a way that makes it easier to spot patterns and relationships.

First, identify what each axis represents; typically, one will show speed (horizontal x-axis) and the other could show distance (vertical y-axis). Identify which lines or curves on the graph represent each method mentioned in the question. Are they straight lines, indicating a constant speed, or curves, showing acceleration or deceleration?

Once you understand what each part of the graph stands for, read the labels carefully to discern units and scales. Graph analysis often involves looking for intersections, slopes, and areas under curves, all of which convey essential information about the relationship between the variables in question. For an ACT Math problem, ensure you're clear on what the graph depicts to make a correct choice among the multiple-choice answers.

Intersection Point

An intersection point on a graph is where two or more lines or curves meet. This point is significant because it represents a set of values that satisfy all equations or conditions represented by these lines. In the context of comparing two methods for calculating distance (D), as in the given exercise, the intersection point indicates the speed where both methods yield the same distance.

To locate the intersection point accurately, trace along the lines until they cross paths. If the graph isn't clearly labeled or the exact point of intersection isn't immediately apparent, you may need to estimate. A common mistake is to misjudge the intersection due to scaling or overcrowding of graph lines, so take your time and analyze the graph closely. The goal is to find the most accurate horizontal coordinate for this intersection point since it will correspond to the initial speed needed to solve the problem.

Comparing Speeds

Comparing speeds in a graph scenario involves matching the visual information with quantitative choices. In our case, the multiple-choice question presents different initial speeds to choose from: 20, 40, 60, and 80 mi/hr. After finding the intersection point on the graph, align it horizontally with the given speeds to see which one it is closest to.

Keep in mind that visual approximation on a graph is not always precise. Therefore, it's necessary to examine the scales of the graph carefully. If the intersection point falls between two values, choose the one it is closest to. This comparison requires a mix of analytical skills and careful observation, ensuring that the graphic representation of speeds aligns with the numerical data of the problem.

Problem Solving Strategy

Every math problem requires a strategy, a clear plan of attack, particularly when solving complex ACT Math problems. Having a strategy in place breaks down the problem into manageable steps, minimizing the chances of feeling overwhelmed.

Start by thoroughly reading the problem to comprehend what is being asked. Then, systematically approach the problem: obtain any necessary figures or data, identify key points such as intersections or pivots in the data, compare these findings to the given options, and draw an evidence-based conclusion.

It is beneficial to stay organized. Jot down important findings as you proceed through each step to avoid confusion. Should the problem involve elements not immediately available (such as a missing figure), consider the steps you would take if you did have the necessary information, or look for alternative ways to reach the solution. The problem-solving strategy is not just about finding an answer, but using logical reasoning, processes, and checks to ensure that the answer makes sense within the context of the given information.