Finite Math Examples

$\frac{1}{2} \cdot ((0.12) ({7.85}^{2})) = (0.12) (9.81) (1.18) + \frac{1}{2} \cdot ((0.12) v^{2})$

Step 1

Rewrite the equation as $(0.12) (9.81) \cdot 1.18 + \frac{1}{2} \cdot (0.12 v^{2}) = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$ .

$(0.12) (9.81) \cdot 1.18 + \frac{1}{2} \cdot (0.12 v^{2}) = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2

Simplify each term.

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Step 2.1

Multiply $(0.12) (9.81) \cdot 1.18$ .

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Step 2.1.1

Multiply $0.12$ by $9.81$ .

$1.1772 \cdot 1.18 + \frac{1}{2} \cdot (0.12 v^{2}) = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2.1.2

Multiply $1.1772$ by $1.18$ .

$1.389096 + \frac{1}{2} \cdot (0.12 v^{2}) = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2.2

Multiply $\frac{1}{2} (0.12 v^{2})$ .

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Step 2.2.1

Combine $0.12$ and $\frac{1}{2}$ .

$1.389096 + \frac{0.12}{2} v^{2} = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2.2.2

Combine $\frac{0.12}{2}$ and $v^{2}$ .

$1.389096 + \frac{0.12 v^{2}}{2} = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2.3

Factor $0.12$ out of $0.12 v^{2}$ .

$1.389096 + \frac{0.12 (v^{2})}{2} = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2.4

Factor $2$ out of $2$ .

$1.389096 + \frac{0.12 (v^{2})}{2 (1)} = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2.5

Separate fractions.

$1.389096 + \frac{0.12}{2} \cdot \frac{v^{2}}{1} = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2.6

Divide $0.12$ by $2$ .

$1.389096 + 0.06 \frac{v^{2}}{1} = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 2.7

Divide $v^{2}$ by $1$ .

$1.389096 + 0.06 v^{2} = \frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$

Step 3

Simplify $\frac{1}{2} \cdot 0.12 \cdot {7.85}^{2}$ .

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Step 3.1

Combine $\frac{1}{2}$ and $0.12$ .

$1.389096 + 0.06 v^{2} = \frac{0.12}{2} \cdot {7.85}^{2}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Divide $0.12$ by $2$ .

$1.389096 + 0.06 v^{2} = 0.06 \cdot {7.85}^{2}$

Step 3.2.2

Raise $7.85$ to the power of $2$ .

$1.389096 + 0.06 v^{2} = 0.06 \cdot 61.6225$

Step 3.2.3

Multiply $0.06$ by $61.6225$ .

$1.389096 + 0.06 v^{2} = 3.69735$

Step 4

Move all terms not containing $v$ to the right side of the equation.

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Step 4.1

Subtract $1.389096$ from both sides of the equation.

$0.06 v^{2} = 3.69735 - 1.389096$

Step 4.2

Subtract $1.389096$ from $3.69735$ .

$0.06 v^{2} = 2.308254$

Step 5

Divide each term in $0.06 v^{2} = 2.308254$ by $0.06$ and simplify.

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Step 5.1

Divide each term in $0.06 v^{2} = 2.308254$ by $0.06$ .

$\frac{0.06 v^{2}}{0.06} = \frac{2.308254}{0.06}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.06$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.06 v^{2}}{0.06} = \frac{2.308254}{0.06}$

Step 5.2.1.2

Divide $v^{2}$ by $1$ .

$v^{2} = \frac{2.308254}{0.06}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $2.308254$ by $0.06$ .

$v^{2} = 38.4709$

Step 6

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

$v = \pm \sqrt{38.4709}$

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.

$v = \sqrt{38.4709}$

Step 7.2

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

$v = - \sqrt{38.4709}$

Step 7.3

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

$v = \sqrt{38.4709}, - \sqrt{38.4709}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$v = \sqrt{38.4709}, - \sqrt{38.4709}$

Decimal Form:

$v = 6.20249143 \dots, - 6.20249143 \dots$