Sure, let's solve the given problems step by step.
### Given:
[tex]\[
\frac{1}{u} + \frac{1}{v} = \frac{1}{f}
\][/tex]
### Part (a): Find [tex]\( u \)[/tex] if [tex]\( v = 6 \)[/tex] and [tex]\( f = 2 \)[/tex]
Given:
- [tex]\( v = 6 \)[/tex]
- [tex]\( f = 2 \)[/tex]
We need to find [tex]\( u \)[/tex].
Substitute the known values into the given equation:
[tex]\[
\frac{1}{u} + \frac{1}{6} = \frac{1}{2}
\][/tex]
Rearrange to isolate [tex]\( \frac{1}{u} \)[/tex]:
[tex]\[
\frac{1}{u} = \frac{1}{2} - \frac{1}{6}
\][/tex]
Find a common denominator:
[tex]\[
\frac{1}{2} = \frac{3}{6}
\][/tex]
So,
[tex]\[
\frac{1}{u} = \frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
\][/tex]
Therefore,
[tex]\[
u = 3
\][/tex]
### Part (b): Find [tex]\( v \)[/tex] if [tex]\( u = 4 \)[/tex] and [tex]\( f = 2 \)[/tex]
Given:
- [tex]\( u = 4 \)[/tex]
- [tex]\( f = 2 \)[/tex]
We need to find [tex]\( v \)[/tex].
Substitute the known values into the given equation:
[tex]\[
\frac{1}{4} + \frac{1}{v} = \frac{1}{2}
\][/tex]
Rearrange to isolate [tex]\( \frac{1}{v} \)[/tex]:
[tex]\[
\frac{1}{v} = \frac{1}{2} - \frac{1}{4}
\][/tex]
Find a common denominator:
[tex]\[
\frac{1}{2} = \frac{2}{4}
\][/tex]
So,
[tex]\[
\frac{1}{v} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}
\][/tex]
Therefore,
[tex]\[
v = 4
\][/tex]
### Part (c): Make [tex]\( f \)[/tex] the subject of the formula
Given:
[tex]\[
\frac{1}{u} + \frac{1}{v} = \frac{1}{f}
\][/tex]
We need to solve for [tex]\( f \)[/tex].
Rearrange the equation:
[tex]\[
\frac{1}{f} = \frac{1}{u} + \frac{1}{v}
\][/tex]
To solve for [tex]\( f \)[/tex], take the reciprocal of both sides:
[tex]\[
f = \frac{1}{\frac{1}{u} + \frac{1}{v}}
\][/tex]
So the formula for [tex]\( f \)[/tex] is:
[tex]\[
f = \frac{1}{\frac{1}{u} + \frac{1}{v}}
\][/tex]
Thus, summarizing all parts:
a. [tex]\( u = 3 \)[/tex] when [tex]\( v = 6 \)[/tex] and [tex]\( f = 2 \)[/tex]
b. [tex]\( v = 4 \)[/tex] when [tex]\( u = 4 \)[/tex] and [tex]\( f = 2 \)[/tex]
c. [tex]\( f = \frac{1}{\frac{1}{u} + \frac{1}{v}} \)[/tex]
