Sure! Let's go through the process step-by-step to solve for [tex]\( c \)[/tex] in the equation [tex]\( 10b = 5(\sqrt{c} + 2) \)[/tex].
1. Isolate [tex]\(\sqrt{c}\)[/tex] on one side of the equation:
Start with the equation:
[tex]\[
10b = 5(\sqrt{c} + 2)
\][/tex]
Divide both sides by 5:
[tex]\[
\frac{10b}{5} = \sqrt{c} + 2
\][/tex]
Simplify:
[tex]\[
2b = \sqrt{c} + 2
\][/tex]
Subtract 2 from both sides to isolate [tex]\(\sqrt{c}\)[/tex]:
[tex]\[
2b - 2 = \sqrt{c}
\][/tex]
2. Square both sides to solve for [tex]\( c \)[/tex]:
[tex]\[
(2b - 2)^2 = c
\][/tex]
Therefore:
[tex]\[
c = (2b - 2)^2
\][/tex]
Given the context of the problem, let's now compare the given potential answers to see which one correctly reflects [tex]\( c = (2b - 2)^2 \)[/tex]:
1. Option [tex]\( c = 10b - 10 - 5 \)[/tex]
[tex]\[
c = 10b - 10 - 5 = 10b - 15
\][/tex]
This does not match our derived equation [tex]\( c = (2b - 2)^2 \)[/tex].
2. Option [tex]\( c = (10b - 10 - 5)^2 \)[/tex]
\]
Simplify inside the parenthesis:
\]
c = (10b - 15)^2
\]
This does not match our derived equation [tex]\( c = (2b - 2)^2 \)[/tex] either.
3. Option [tex]\( c = \frac{(10b - 2)^2}{25} \)[/tex]
Let's simplify it:
[tex]\[
c = \frac{(10b - 2)^2}{25}
\][/tex]
\)
This doesn't simplify correctly to match [tex]\( c = (2b - 2)^2 \)[/tex].
Given all of the options provided, none exactly match with the derived formula for [tex]\( c \)[/tex].
Therefore, the correct solution, based on the initial derivation, is:
[tex]\[
c = (2b - 2)^2
\][/tex]
This detailed step-by-step solution shows the method to derive [tex]\( c \)[/tex] from the given equation.
