Pre-Algebra Examples

 $\begin{array}{r} h = & 6 m \\ l = & 10 m \\ w = & x \end{array}$

Step 1

The surface area of a pyramid is equal to the sum of the areas of each side of the pyramid. The base of the pyramid has area $l w$ , and $s_{l}$ and $s_{w}$ represent the slant height on the length and slant height on the width.

$(l e n g t h) \cdot (w i d t h) + (w i d t h) \cdot s_{l} + (l e n g t h) \cdot s_{w}$

Step 2

Substitute the values of the length $l = 10$ , the width $w = x$ , and the height $h = 6$ into the formula for surface area of a pyramid.

$10 \cdot x + x \cdot \sqrt{{(\frac{10}{2})}^{2} + {(6)}^{2}} + 10 \cdot \sqrt{{(\frac{x}{2})}^{2} + {(6)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Divide $10$ by $2$ .

$10 x + x \cdot \sqrt{5^{2} + {(6)}^{2}} + 10 \cdot \sqrt{{(\frac{x}{2})}^{2} + {(6)}^{2}}$

Step 3.2

Raise $5$ to the power of $2$ .

$10 x + x \cdot \sqrt{25 + {(6)}^{2}} + 10 \cdot \sqrt{{(\frac{x}{2})}^{2} + {(6)}^{2}}$

Step 3.3

Raise $6$ to the power of $2$ .

$10 x + x \cdot \sqrt{25 + 36} + 10 \cdot \sqrt{{(\frac{x}{2})}^{2} + {(6)}^{2}}$

Step 3.4

Add $25$ and $36$ .

$10 x + x \cdot \sqrt{61} + 10 \cdot \sqrt{{(\frac{x}{2})}^{2} + {(6)}^{2}}$

Step 3.5

Apply the product rule to $\frac{x}{2}$ .

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{x^{2}}{2^{2}} + {(6)}^{2}}$

Step 3.6

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

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{x^{2}}{4} + {(6)}^{2}}$

Step 3.7

Raise $6$ to the power of $2$ .

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{x^{2}}{4} + 36}$

Step 3.8

To write $36$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{x^{2}}{4} + 36 \cdot \frac{4}{4}}$

Step 3.9

Combine $36$ and $\frac{4}{4}$ .

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{x^{2}}{4} + \frac{36 \cdot 4}{4}}$

Step 3.10

Combine the numerators over the common denominator.

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{x^{2} + 36 \cdot 4}{4}}$

Step 3.11

Multiply $36$ by $4$ .

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{x^{2} + 144}{4}}$

Step 3.12

Rewrite $\frac{x^{2} + 144}{4}$ as ${(\frac{1}{2})}^{2} (x^{2} + 144)$ .

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Step 3.12.1

Factor the perfect power $1^{2}$ out of $x^{2} + 144$ .

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{1^{2} (x^{2} + 144)}{4}}$

Step 3.12.2

Factor the perfect power $2^{2}$ out of $4$ .

$10 x + x \sqrt{61} + 10 \cdot \sqrt{\frac{1^{2} (x^{2} + 144)}{2^{2} \cdot 1}}$

Step 3.12.3

Rearrange the fraction $\frac{1^{2} (x^{2} + 144)}{2^{2} \cdot 1}$ .

$10 x + x \sqrt{61} + 10 \cdot \sqrt{{(\frac{1}{2})}^{2} (x^{2} + 144)}$

Step 3.13

Pull terms out from under the radical.

$10 x + x \sqrt{61} + 10 \cdot (\frac{1}{2} \sqrt{x^{2} + 144})$

Step 3.14

Combine $\frac{1}{2}$ and $\sqrt{x^{2} + 144}$ .

$10 x + x \sqrt{61} + 10 \cdot \frac{\sqrt{x^{2} + 144}}{2}$

Step 3.15

Cancel the common factor of $2$ .

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Step 3.15.1

Factor $2$ out of $10$ .

$10 x + x \sqrt{61} + 2 (5) \cdot \frac{\sqrt{x^{2} + 144}}{2}$

Step 3.15.2

Cancel the common factor.

$10 x + x \sqrt{61} + 2 \cdot 5 \cdot \frac{\sqrt{x^{2} + 144}}{2}$

Step 3.15.3

Rewrite the expression.

$10 x + x \sqrt{61} + 5 \cdot \sqrt{x^{2} + 144}$

$10 x + x \sqrt{61} + 5 \sqrt{x^{2} + 144}$

$10 x + x \sqrt{61} + 5 \sqrt{x^{2} + 144} m^{2}$