Basic Math Examples

 $\begin{array}{r} h = & 8 \\ l = & 2 \\ w = & 5 \end{array}$

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 $l = 2$ , the width $w = 5$ , and the height $h = 8$ into the formula for surface area of a pyramid.

$2 \cdot 5 + 5 \cdot \sqrt{{(\frac{2}{2})}^{2} + {(8)}^{2}} + 2 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Multiply $2$ by $5$ .

$10 + 5 \cdot \sqrt{{(\frac{2}{2})}^{2} + {(8)}^{2}} + 2 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Step 3.2

Divide $2$ by $2$ .

$10 + 5 \cdot \sqrt{1^{2} + {(8)}^{2}} + 2 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Step 3.3

One to any power is one.

$10 + 5 \cdot \sqrt{1 + {(8)}^{2}} + 2 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Step 3.4

Raise $8$ to the power of $2$ .

$10 + 5 \cdot \sqrt{1 + 64} + 2 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Step 3.5

Add $1$ and $64$ .

$10 + 5 \cdot \sqrt{65} + 2 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Step 3.6

Apply the product rule to $\frac{5}{2}$ .

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{5^{2}}{2^{2}} + {(8)}^{2}}$

Step 3.7

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

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{25}{2^{2}} + {(8)}^{2}}$

Step 3.8

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

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{25}{4} + {(8)}^{2}}$

Step 3.9

Raise $8$ to the power of $2$ .

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{25}{4} + 64}$

Step 3.10

To write $64$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{25}{4} + 64 \cdot \frac{4}{4}}$

Step 3.11

Combine $64$ and $\frac{4}{4}$ .

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{25}{4} + \frac{64 \cdot 4}{4}}$

Step 3.12

Combine the numerators over the common denominator.

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{25 + 64 \cdot 4}{4}}$

Step 3.13

Simplify the numerator.

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Step 3.13.1

Multiply $64$ by $4$ .

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{25 + 256}{4}}$

Step 3.13.2

Add $25$ and $256$ .

$10 + 5 \sqrt{65} + 2 \cdot \sqrt{\frac{281}{4}}$

Step 3.14

Rewrite $\sqrt{\frac{281}{4}}$ as $\frac{\sqrt{281}}{\sqrt{4}}$ .

$10 + 5 \sqrt{65} + 2 \cdot \frac{\sqrt{281}}{\sqrt{4}}$

Step 3.15

Simplify the denominator.

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Step 3.15.1

Rewrite $4$ as $2^{2}$ .

$10 + 5 \sqrt{65} + 2 \cdot \frac{\sqrt{281}}{\sqrt{2^{2}}}$

Step 3.15.2

Pull terms out from under the radical, assuming positive real numbers.

$10 + 5 \sqrt{65} + 2 \cdot \frac{\sqrt{281}}{2}$

Step 3.16

Cancel the common factor of $2$ .

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Step 3.16.1

Cancel the common factor.

$10 + 5 \sqrt{65} + 2 \cdot \frac{\sqrt{281}}{2}$

Step 3.16.2

Rewrite the expression.

$10 + 5 \sqrt{65} + \sqrt{281}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$67.0743$

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