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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 below:

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 :

The student's answer is actually incorrect. Let's analyze the student's work step-by-step and identify the correct approach.

1. Total Sections Calculation:

To partition the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in a ratio of [tex]\( 3:4 \)[/tex], we first need to determine the total number of sections:
[tex]\[ \text{total sections} = 3 + 4 = 7 \][/tex]

2. Fraction for the Partition:

We then use the fraction that corresponds to the given ratio. Since we want the point [tex]\( C \)[/tex] such that it partitions the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 3:4 \)[/tex]:
[tex]\[ \text{fraction} = \frac{3}{7} \][/tex]

3. Calculate the Coordinate of Point [tex]\( C \)[/tex]:

The formula to find the coordinate of point [tex]\( C \)[/tex] is given by:
[tex]\[ C = \left( \frac{3}{7} \right)(B - A) + A \][/tex]

Plugging in the given values [tex]\( A = -6 \)[/tex] and [tex]\( B = 2 \)[/tex]:
[tex]\[ C = \left( \frac{3}{7} \right)(2 - (-6)) + (-6) \][/tex]
[tex]\[ C = \left( \frac{3}{7} \right)(8) + (-6) \][/tex]
[tex]\[ C = \left( \frac{24}{7} \right) - 6 \][/tex]
[tex]\[ C = 3.4285714285714284 - 6 \][/tex]
[tex]\[ C = -2.5714285714285716 \][/tex]

4. Verification:

The correct position of point [tex]\( C \)[/tex] is approximately [tex]\(-2.57\)[/tex], not 0.

Thus, the student's error lies in the initial application of the ratio. They mistakenly used the fraction [tex]\( \frac{3}{4} \)[/tex] of the length instead of the correct fraction [tex]\( \frac{3}{7} \)[/tex].

So, the correct answer to your question is:
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].
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