Calculus Examples

$C = \frac{160}{v} + \frac{V^{2}}{100}$

Step 1

Rewrite the equation as $\frac{160}{v} + \frac{V^{2}}{100} = C$ .

$\frac{160}{v} + \frac{V^{2}}{100} = C$

Step 2

Subtract $\frac{160}{v}$ from both sides of the equation.

$\frac{V^{2}}{100} = C - \frac{160}{v}$

Step 3

Multiply both sides of the equation by $100$ .

$100 \frac{V^{2}}{100} = 100 (C - \frac{160}{v})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $100$ .

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Step 4.1.1.1

Cancel the common factor.

$100 \frac{V^{2}}{100} = 100 (C - \frac{160}{v})$

Step 4.1.1.2

Rewrite the expression.

$V^{2} = 100 (C - \frac{160}{v})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $100 (C - \frac{160}{v})$ .

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Step 4.2.1.1

Apply the distributive property.

$V^{2} = 100 C + 100 (- \frac{160}{v})$

Step 4.2.1.2

Multiply $100 (- \frac{160}{v})$ .

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Step 4.2.1.2.1

Multiply $- 1$ by $100$ .

$V^{2} = 100 C - 100 \frac{160}{v}$

Step 4.2.1.2.2

Combine $- 100$ and $\frac{160}{v}$ .

$V^{2} = 100 C + \frac{- 100 \cdot 160}{v}$

Step 4.2.1.2.3

Multiply $- 100$ by $160$ .

$V^{2} = 100 C + \frac{- 16000}{v}$

Step 4.2.1.3

Move the negative in front of the fraction.

$V^{2} = 100 C - \frac{16000}{v}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$V = \pm \sqrt{100 C - \frac{16000}{v}}$

Step 6

Simplify $\pm \sqrt{100 C - \frac{16000}{v}}$ .

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Step 6.1

Factor $100$ out of $100 C - \frac{16000}{v}$ .

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Step 6.1.1

Factor $100$ out of $100 C$ .

$V = \pm \sqrt{100 (C) - \frac{16000}{v}}$

Step 6.1.2

Factor $100$ out of $- \frac{16000}{v}$ .

$V = \pm \sqrt{100 (C) + 100 (- \frac{160}{v})}$

Step 6.1.3

Factor $100$ out of $100 (C) + 100 (- \frac{160}{v})$ .

$V = \pm \sqrt{100 (C - \frac{160}{v})}$

Step 6.2

To write $C$ as a fraction with a common denominator, multiply by $\frac{v}{v}$ .

$V = \pm \sqrt{100 (\frac{C v}{v} - \frac{160}{v})}$

Step 6.3

Combine the numerators over the common denominator.

$V = \pm \sqrt{100 \frac{C v - 160}{v}}$

Step 6.4

Combine $100$ and $\frac{C v - 160}{v}$ .

$V = \pm \sqrt{\frac{100 (C v - 160)}{v}}$

Step 6.5

Rewrite $\sqrt{\frac{100 (C v - 160)}{v}}$ as $\frac{\sqrt{100 (C v - 160)}}{\sqrt{v}}$ .

$V = \pm \frac{\sqrt{100 (C v - 160)}}{\sqrt{v}}$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Rewrite $100$ as $10^{2}$ .

$V = \pm \frac{\sqrt{10^{2} (C v - 160)}}{\sqrt{v}}$

Step 6.6.2

Pull terms out from under the radical.

$V = \pm \frac{10 \sqrt{C v - 160}}{\sqrt{v}}$

Step 6.7

Multiply $\frac{10 \sqrt{C v - 160}}{\sqrt{v}}$ by $\frac{\sqrt{v}}{\sqrt{v}}$ .

$V = \pm \frac{10 \sqrt{C v - 160}}{\sqrt{v}} \cdot \frac{\sqrt{v}}{\sqrt{v}}$

Step 6.8

Combine and simplify the denominator.

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Step 6.8.1

Multiply $\frac{10 \sqrt{C v - 160}}{\sqrt{v}}$ by $\frac{\sqrt{v}}{\sqrt{v}}$ .

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{\sqrt{v} \sqrt{v}}$

Step 6.8.2

Raise $\sqrt{v}$ to the power of $1$ .

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{{\sqrt{v}}^{1} \sqrt{v}}$

Step 6.8.3

Raise $\sqrt{v}$ to the power of $1$ .

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{{\sqrt{v}}^{1} {\sqrt{v}}^{1}}$

Step 6.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{{\sqrt{v}}^{1 + 1}}$

Step 6.8.5

Add $1$ and $1$ .

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{{\sqrt{v}}^{2}}$

Step 6.8.6

Rewrite ${\sqrt{v}}^{2}$ as $v$ .

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Step 6.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{v}$ as $v^{\frac{1}{2}}$ .

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{{(v^{\frac{1}{2}})}^{2}}$

Step 6.8.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{v^{\frac{1}{2} \cdot 2}}$

Step 6.8.6.3

Combine $\frac{1}{2}$ and $2$ .

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{v^{\frac{2}{2}}}$

Step 6.8.6.4

Cancel the common factor of $2$ .

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Step 6.8.6.4.1

Cancel the common factor.

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{v^{\frac{2}{2}}}$

Step 6.8.6.4.2

Rewrite the expression.

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{v^{1}}$

Step 6.8.6.5

Simplify.

$V = \pm \frac{10 \sqrt{C v - 160} \sqrt{v}}{v}$

Step 6.9

Combine using the product rule for radicals.

$V = \pm \frac{10 \sqrt{v (C v - 160)}}{v}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$V = \frac{10 \sqrt{v (C v - 160)}}{v}$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$V = - \frac{10 \sqrt{v (C v - 160)}}{v}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$V = \frac{10 \sqrt{v (C v - 160)}}{v}$

$V = - \frac{10 \sqrt{v (C v - 160)}}{v}$

$V = \frac{10 \sqrt{v (C v - 160)}}{v}$

$V = - \frac{10 \sqrt{v (C v - 160)}}{v}$