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Rewrite the task so that it makes sense, correct any grammar and spelling errors, and format it properly:
```
Given the equation:
[tex]\[ \left(p+\frac{a}{v^2}\right)(v-b)=\text{constant} \][/tex]

If [tex]\( p \)[/tex] is the pressure and [tex]\( V \)[/tex] is the volume, what are the units of [tex]\( a \)[/tex]?
```



Answer :

GinnyAnswer
Certainly! Let's start by looking at the equation given:

[tex]\[ \left(p + \frac{a}{v^2}\right)(v - b) = \text{constant} \][/tex]

We need to solve the equation for [tex]\( p \)[/tex].

### Step-by-Step Solution:

1. Expand the Given Equation:

First, let's expand the left side of the equation:

[tex]\[ \left(p + \frac{a}{v^2}\right)(v - b) = \text{constant} \][/tex]

[tex]\[ p(v - b) + \frac{a}{v^2}(v - b) = \text{constant} \][/tex]

2. Distribute Each Term:

Now distribute [tex]\( p \)[/tex] and [tex]\(\frac{a}{v^2}\)[/tex] across [tex]\( (v - b) \)[/tex]:

[tex]\[ p \cdot v - p \cdot b + \frac{a \cdot v}{v^2} - \frac{a \cdot b}{v^2} = \text{constant} \][/tex]

Simplify the terms:

[tex]\[ p \cdot v - p \cdot b + \frac{a}{v} - \frac{a \cdot b}{v^2} = \text{constant} \][/tex]

3. Isolate the [tex]\( p \)[/tex] Terms:

We need to isolate [tex]\( p \)[/tex] on one side of the equation. So let's collect all [tex]\( p \)[/tex]-related terms on one side:

[tex]\[ p(v - b) = \text{constant} - \frac{a}{v} + \frac{a \cdot b}{v^2} \][/tex]

4. Solve for [tex]\( p \)[/tex]:

Divide both sides of the equation by [tex]\((v - b)\)[/tex] to solve for [tex]\( p \)[/tex]:

[tex]\[ p = \frac{\text{constant} - \frac{a}{v} + \frac{a \cdot b}{v^2}}{v - b} \][/tex]

5. Combine the Terms:

Combine the terms in the numerator into a single fraction:

[tex]\[ p = \frac{\text{constant}v^2 - av + ab}{v^2(v - b)} \][/tex]

Since we need to represent it cleanly, let’s reformat:

[tex]\[ p = \frac{-a \cdot b + a \cdot v - \text{constant} \cdot v^2}{v^2 (b - v)} \][/tex]

Since dividing by [tex]\((b - v)\)[/tex] is the same as flipping the signs in the numerator, we have:

[tex]\[ p = \frac{-a \cdot b + a \cdot v - \text{constant} \cdot v^2}{v^2 (b - v)} \][/tex]

This is a detailed step-by-step solution for isolating [tex]\( p \)[/tex] in the provided equation.

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