Sure! Let's solve the problem step-by-step.
We are given:
- Point [tex]\( A \)[/tex] at 6 on the number line.
- Point [tex]\( C \)[/tex] at 1.875 on the number line.
- The ratio [tex]\( AC : CB = 3 : 5 \)[/tex].
First, we need to calculate the length of [tex]\( AC \)[/tex]:
[tex]\[ AC = |6 - 1.875| = 4.125 \][/tex]
Let [tex]\( x \)[/tex] be the length of [tex]\( CB \)[/tex].
Since [tex]\( AC \)[/tex] and [tex]\( CB \)[/tex] are in the ratio [tex]\( 3:5 \)[/tex], we can write:
[tex]\[ \frac{AC}{CB} = \frac{3}{5} \][/tex]
In this ratio, let's use the known values:
[tex]\[ \frac{4.125}{x} = \frac{3}{5} \][/tex]
To find [tex]\( x \)[/tex], we solve for [tex]\( x \)[/tex]:
[tex]\[ 4.125 \cdot 5 = 3 \cdot x \][/tex]
[tex]\[ 20.625 = 3x \][/tex]
[tex]\[ x = \frac{20.625}{3} \][/tex]
[tex]\[ x = 6.875 \][/tex]
So, the length of [tex]\( CB \)[/tex] is 6.875 units.
Next, we need to calculate the total length of [tex]\( \overline{AB} \)[/tex]:
[tex]\[ AB = AC + CB \][/tex]
[tex]\[ AB = 4.125 + 6.875 \][/tex]
[tex]\[ AB = 11 \][/tex]
Therefore, the length of [tex]\( \overline{AB} \)[/tex] is 11 units.
The correct answer is:
C. [tex]\( AB = 11 \)[/tex] units
