Calculus Examples

$215 = \frac{1}{2} \cdot (a {(\frac{65}{a})}^{2})$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (a {(\frac{65}{a})}^{2}) = 215$ .

$\frac{1}{2} \cdot (a {(\frac{65}{a})}^{2}) = 215$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (a {(\frac{65}{a})}^{2})) = 2 \cdot 215$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $2 (\frac{1}{2} \cdot (a {(\frac{65}{a})}^{2}))$ .

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Step 3.1.1.1

Simplify the expression.

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Step 3.1.1.1.1

Apply the product rule to $\frac{65}{a}$ .

$2 (\frac{1}{2} \cdot (a \frac{65^{2}}{a^{2}})) = 2 \cdot 215$

Step 3.1.1.1.2

Raise $65$ to the power of $2$ .

$2 (\frac{1}{2} \cdot (a \frac{4225}{a^{2}})) = 2 \cdot 215$

Step 3.1.1.2

Cancel the common factor of $a$ .

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Step 3.1.1.2.1

Factor $a$ out of $a^{2}$ .

$2 (\frac{1}{2} \cdot (a \frac{4225}{a \cdot a})) = 2 \cdot 215$

Step 3.1.1.2.2

Cancel the common factor.

$2 (\frac{1}{2} \cdot (a \frac{4225}{a \cdot a})) = 2 \cdot 215$

Step 3.1.1.2.3

Rewrite the expression.

$2 (\frac{1}{2} \cdot \frac{4225}{a}) = 2 \cdot 215$

Step 3.1.1.3

Multiply $\frac{1}{2}$ by $\frac{4225}{a}$ .

$2 \frac{4225}{2 a} = 2 \cdot 215$

Step 3.1.1.4

Cancel the common factor of $2$ .

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Step 3.1.1.4.1

Factor $2$ out of $2 a$ .

$2 \frac{4225}{2 (a)} = 2 \cdot 215$

Step 3.1.1.4.2

Cancel the common factor.

$2 \frac{4225}{2 a} = 2 \cdot 215$

Step 3.1.1.4.3

Rewrite the expression.

$\frac{4225}{a} = 2 \cdot 215$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $215$ .

$\frac{4225}{a} = 430$

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$a, 1$

Step 4.2

The LCM of one and any expression is the expression.

$a$

Step 5

Multiply each term in $\frac{4225}{a} = 430$ by $a$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{4225}{a} = 430$ by $a$ .

$\frac{4225}{a} a = 430 a$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $a$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4225}{a} a = 430 a$

Step 5.2.1.2

Rewrite the expression.

$4225 = 430 a$

Step 6

Solve the equation.

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Step 6.1

Rewrite the equation as $430 a = 4225$ .

$430 a = 4225$

Step 6.2

Divide each term in $430 a = 4225$ by $430$ and simplify.

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Step 6.2.1

Divide each term in $430 a = 4225$ by $430$ .

$\frac{430 a}{430} = \frac{4225}{430}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $430$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{430 a}{430} = \frac{4225}{430}$

Step 6.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{4225}{430}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Cancel the common factor of $4225$ and $430$ .

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Step 6.2.3.1.1

Factor $5$ out of $4225$ .

$a = \frac{5 (845)}{430}$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $5$ out of $430$ .

$a = \frac{5 \cdot 845}{5 \cdot 86}$

Step 6.2.3.1.2.2

Cancel the common factor.

$a = \frac{5 \cdot 845}{5 \cdot 86}$

Step 6.2.3.1.2.3

Rewrite the expression.

$a = \frac{845}{86}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{845}{86}$

Decimal Form:

$a = 9.82558139 \dots$

Mixed Number Form:

$a = 9 \frac{71}{86}$