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A student wants to find point [tex]\( C \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] on a number line such that the segment is partitioned in a ratio of [tex]\( 3:4 \)[/tex]. Point [tex]\( A \)[/tex] is at [tex]\(-6\)[/tex] and point [tex]\( B \)[/tex] is at [tex]\( 2 \)[/tex]. The student's work is shown.

1. [tex]\( c = \left(\frac{3}{4}\right)(2 - (-6)) + (-6) \)[/tex]
2. [tex]\( c = \left(\frac{3}{4}\right)(8) - 6 \)[/tex]
3. [tex]\( c = 6 - 6 \)[/tex]
4. [tex]\( c = 0 \)[/tex]

Analyze the student's work. Is the answer correct? Explain.

A. No, the student should have added [tex]\( 3 + 4 \)[/tex] to get the total number of sections and used the fraction [tex]\(\frac{3}{7}\)[/tex] instead of [tex]\(\frac{3}{4}\)[/tex].

B. No, the student should have subtracted [tex]\( 2 \)[/tex] from [tex]\(-6\)[/tex] to find the distance.

C. No, the student should have added [tex]\( 2 \)[/tex] at the end to add to the starting point.

D. Yes, the student's answer is correct.

Sagot :

GinnyAnswer
The student's methodology contains an error. Let's go through the process step-by-step to identify where the mistake lies.

1. Identifying the Points and the Total Number of Sections:
- Point [tex]\( A \)[/tex] is at -6 and Point [tex]\( B \)[/tex] is at 2.
- We need to partition the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 3:4 \)[/tex].
- The total number of sections created by this ratio is [tex]\( 3 + 4 = 7 \)[/tex].

2. Calculating the Specific Fraction of the Segment:
- The correct fraction that represents the position of point [tex]\( C \)[/tex] along the line segment [tex]\( AB \)[/tex] is [tex]\( \frac{3}{7} \)[/tex]. This fraction signifies that point [tex]\( C \)[/tex] is closer to [tex]\( A \)[/tex] than [tex]\( B \)[/tex], covering 3 parts out of the total 7 parts of the segment.

3. Finding the Distance Covered by the Fraction:
- Compute the distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]: since [tex]\( B - A = 2 - (-6) = 8 \)[/tex].
- The scaled distance for point [tex]\( C \)[/tex] using the fraction [tex]\( \frac{3}{7} \)[/tex] is calculated as [tex]\( \frac{3}{7} \times 8 = 3.4285714285714284 \)[/tex].

4. Determining the Position of Point [tex]\( C \)[/tex]:
- To find [tex]\( C \)[/tex], start at [tex]\( A \)[/tex] and move the scaled distance towards [tex]\( B \)[/tex]:
- Thus, point [tex]\( C \)[/tex] is located at [tex]\( A + \text{(fraction of the distance)} \)[/tex], which is [tex]\( -6 + \frac{3}{7} \times 8 \)[/tex].
- This simplifies to:
[tex]\[ -6 + 3.4285714285714284 = -2.5714285714285716 \][/tex]

Therefore, point [tex]\( C \)[/tex] is at approximately [tex]\(-2.57\)[/tex].

Analysis of the Student's Work:

- The student mistakenly used [tex]\( \frac{3}{4} \)[/tex] instead of [tex]\( \frac{3}{7} \)[/tex].
- Correcting the student's fractions and calculations reveals the accurate placement of [tex]\( C \)[/tex].

Thus, the student's approach is incorrect and the accurate calculations show that point [tex]\( C \)[/tex] is [tex]\( -2.57 \)[/tex].
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