To make [tex]\( x \)[/tex] the subject of the formula [tex]\( S = W \sqrt{a^2 - x^2} \)[/tex], follow these steps:
### Step 1: Rearrange for [tex]\( W \)[/tex]
Start with the given equation:
[tex]\[ S = W \sqrt{a^2 - x^2} \][/tex]
We first need to isolate [tex]\( W \)[/tex] on one side.
### Step 2: Solve for [tex]\( W \)[/tex]
Divide both sides by [tex]\( \sqrt{a^2 - x^2} \)[/tex]:
[tex]\[ W = \frac{S}{\sqrt{a^2 - x^2}} \][/tex]
### Step 3: Apply for [tex]\( x = 9 \)[/tex]
Next, let's apply this to the case when [tex]\( x = 9 \)[/tex]:
[tex]\[ W = \frac{S}{\sqrt{a^2 - 9^2}} \][/tex]
[tex]\[ W = \frac{S}{\sqrt{a^2 - 81}} \][/tex]
### Step 4: Use the value [tex]\( W \)[/tex] in the equation for [tex]\( x = 6 \)[/tex]
Now, we know that [tex]\( W = \frac{S}{\sqrt{a^2 - 81}} \)[/tex]. Next, substitute this into the original equation for the second case when [tex]\( x = 6 \)[/tex]:
[tex]\[ S = W \sqrt{a^2 - 6^2} \][/tex]
[tex]\[ S = W \sqrt{a^2 - 36} \][/tex]
Substitute the value of [tex]\( W \)[/tex]:
[tex]\[ S = \left(\frac{S}{\sqrt{a^2 - 81}}\right) \sqrt{a^2 - 36} \][/tex]
### Step 5: Simplify the equation
Simplify the equation by multiplying both sides by [tex]\( \sqrt{a^2 - 81} \)[/tex] to clear the denominator:
[tex]\[ S \cdot \sqrt{a^2 - 81} = S \cdot \sqrt{a^2 - 36} \][/tex]
This simplifies to:
[tex]\[ \sqrt{a^2 - 81} = \sqrt{a^2 - 36} \][/tex]
### Conclusion
The steps show that the original relationships hold true when substituting values for [tex]\( x \)[/tex]. The solutions and equation:
[tex]\[ W = \frac{S}{\sqrt{a^2 - 81}} \][/tex]
[tex]\[ S = S \cdot \frac{\sqrt{a^2 - 36}}{\sqrt{a^2 - 81}} \][/tex]
confirm that the equation remains consistent when substituting the value of [tex]\( W \)[/tex].
