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Sagot :
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]
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]
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