To determine the correct proportion to solve for [tex]\(x\)[/tex], let's examine each option individually to see which one will give us a consistent solution:
### Option A: [tex]\(\frac{64}{x} = \frac{x}{36}\)[/tex]
1. Multiply both sides by [tex]\(x \cdot 36\)[/tex]:
[tex]\[
64 \cdot 36 = x \cdot x \implies 2304 = x^2
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \sqrt{2304} \implies x = \pm 48
\][/tex]
Thus, the solutions are [tex]\( x = -48 \)[/tex] and [tex]\( x = 48 \)[/tex].
### Option B: [tex]\(\frac{x}{64} = \frac{64}{36}\)[/tex]
1. Multiply both sides by [tex]\(64 \cdot 36\)[/tex]:
[tex]\[
x \cdot 36 = 64 \cdot 64 \implies 36x = 4096
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{4096}{36} \implies x \approx 113.78
\][/tex]
### Option C: [tex]\(\frac{x}{36} = \frac{100}{x}\)[/tex]
1. Multiply both sides by [tex]\(36 \cdot x\)[/tex]:
[tex]\[
x^2 = 100 \cdot 36 \implies x^2 = 3600
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \sqrt{3600} \implies x = \pm 60
\][/tex]
Thus, the solutions are [tex]\( x = -60 \)[/tex] and [tex]\( x = 60 \)[/tex].
### Option D: [tex]\(\frac{x}{100} = \frac{36}{64}\)[/tex]
1. Multiply both sides by [tex]\(100 \cdot 64\)[/tex]:
[tex]\[
x \cdot 64 = 36 \cdot 100 \implies 64x = 3600
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{3600}{64} \implies x = 56.25
\][/tex]
### Conclusion
Now we check the validity of the solutions:
- Option A yields [tex]\( x = -48 \)[/tex] and [tex]\( x = 48 \)[/tex]
- Option B yields [tex]\( x \approx 113.78 \)[/tex]
- Option C yields [tex]\( x = -60 \)[/tex] and [tex]\( x = 60 \)[/tex]
- Option D yields [tex]\( x = 56.25 \)[/tex]
Given that 113.78 is a clear specific value not arriving from square roots, fraction approximations, or factoring out negatives, it is likely a correct choice. Therefore, the correct proportion for solving [tex]\( x \)[/tex] is given by Option B:
[tex]\[
\boxed{\frac{x}{64} = \frac{64}{36}}
\][/tex]
