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solve for r divided by x, and keep r and x to one side of the equation

Let's start by analyzing and simplifying the given equation:

\[
D = l \times b \times \left(1.25 \times (r \times l) + \phi + (x \times D) \times \sin\left(\cos^{-1}(\phi)\right)\right)
\]

We are asked to solve for \( \frac{r}{x} \), and rearrange the equation to keep \( r \) and \( x \) on one side.

### Step 1: Expand the equation
First, let's expand the terms inside the parentheses:

\[
D = l \times b \times \left(1.25 \times r \times l + \phi + x \times D \times \sin\left(\cos^{-1}(\phi)\right)\right)
\]

### Step 2: Distribute the \( l \times b \)
Distribute \( l \times b \) across each term inside the parentheses:

\[
D = l \times b \times 1.25 \times r \times l + l \times b \times \phi + l \times b \times x \times D \times \sin\left(\cos^{-1}(\phi)\right)
\]

\[
D = 1.25 \times l^2 \times b \times r + l \times b \times \phi + l \times b \times x \times D \times \sin\left(\cos^{-1}(\phi)\right)
\]

### Step 3: Isolate the terms with \( r \) and \( x \)
Move the term containing \( x \times D \) to the other side of the equation:

\[
D - l \times b \times x \times D \times \sin\left(\cos^{-1}(\phi)\right) = 1.25 \times l^2 \times b \times r + l \times b \times \phi
\]

### Step 4: Factor out \( D \) on the left side
Factor \( D \) out of the left-hand side of the equation:

\[
D \times \left(1 - l \times b \times x \times \sin\left(\cos^{-1}(\phi)\right)\right) = 1.25 \times l^2 \times b \times r + l \times b \times \phi
\]

### Step 5: Solve for \( r \) divided by \( x \)
Now, divide both sides by \( 1.25 \times l^2 \times b \) to isolate \( r \):

\[
\frac{r}{x} = \frac{D \times \left(1 - l \times b \times x \times \sin\left(\cos^{-1}(\phi)\right)\right) - l \times b \times \phi}{1.25 \times l^2 \times b \times x}
\]

This is the expression for \( \frac{r}{x} \) with \( r \) and \( x \) kept on one side of the equation.

Do you want further details, or do you have any questions?

Here are some related questions you might find interesting:
1. How does the value of \( \phi \) affect the equation?
2. What happens if \( \sin(\cos^{-1}(\phi)) \) equals zero?
3. How can we simplify this equation for specific values of \( l \), \( b \), and \( \phi \)?
4. What if we need to solve for \( r \) alone, without the division by \( x \)?
5. Can we interpret \( \cos^{-1}(\phi) \) in a different way, perhaps using trigonometric identities?
6. What are the possible values for \( \phi \) that make the equation valid?
7. How does changing \( D \) influence the relationship between \( r \) and \( x \)?
8. How would the equation change if \( r \) or \( x \) were negative?

**Tip:** When solving equations, try to isolate the variable of interest early in the process, which can help reduce errors and simplify calculations.

Explore how to solve for r divided by x in a complex equation step-by-step. This detailed analysis covers trigonometric identities and equation manipulation. Suitable for advanced mathematics students and professionals.

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