To solve for [tex]\( x \)[/tex] in a proportion, we need to analyze each given option and find which proportion results in a reasonable value for [tex]\( x \)[/tex].
Let's examine each option step-by-step:
### Option a: [tex]\(\frac{64}{x} = \frac{x}{36}\)[/tex]
1. Set the cross products equal:
[tex]\[
64 \cdot 36 = x \cdot x
\][/tex]
2. Simplify the equation:
[tex]\[
2304 = x^2
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{2304}
\][/tex]
4. Calculate the square root:
[tex]\[
x = 48
\][/tex]
So, [tex]\( x = 48 \)[/tex] is a consistent result.
### Option b: [tex]\(\frac{x}{64} = \frac{64}{36}\)[/tex]
1. Set the cross products equal:
[tex]\[
x \cdot 36 = 64 \cdot 64
\][/tex]
2. Simplify the equation:
[tex]\[
36x = 4096
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{4096}{36}
\][/tex]
4. Simplify the fraction:
[tex]\[
x \approx 113.78
\][/tex]
This result seems less likely for a simple proportion problem compared to option a.
### Option c: [tex]\(\frac{x}{36} = \frac{100}{x}\)[/tex]
1. Set the cross products equal:
[tex]\[
x \cdot x = 36 \cdot 100
\][/tex]
2. Simplify the equation:
[tex]\[
x^2 = 3600
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{3600}
\][/tex]
4. Calculate the square root:
[tex]\[
x = 60
\][/tex]
This also results in a reasonable value for [tex]\( x \)[/tex].
### Option d: [tex]\(\frac{x}{100} = \frac{36}{64}\)[/tex]
1. Set the cross products equal:
[tex]\[
x \cdot 64 = 36 \cdot 100
\][/tex]
2. Simplify the equation:
[tex]\[
64x = 3600
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3600}{64}
\][/tex]
4. Simplify the fraction:
[tex]\[
x = 56.25
\][/tex]
The result [tex]\( x = 56.25 \)[/tex] is less likely compared to options a and c in terms of simplicity and common reasonable values.
### Conclusion
After analyzing all options, the most consistent and straightforward result aligns with option (a) [tex]\(\frac{64}{x} = \frac{x}{36}\)[/tex]. Therefore, the correct proportion to use for solving for [tex]\( x \)[/tex] is:
[tex]\[
\boxed{\frac{64}{x}=\frac{x}{36}}
\][/tex]
