Pre-Algebra Examples

 $\begin{array}{r} h = & 10.8 \\ l = & 10 \\ w = & 5 \end{array}$

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 $l = 10$ , the width $w = 5$ , and the height $h = 10.8$ into the formula for surface area of a pyramid.

$10 \cdot 5 + 5 \cdot \sqrt{{(\frac{10}{2})}^{2} + {(10.8)}^{2}} + 10 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(10.8)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Multiply $10$ by $5$ .

$50 + 5 \cdot \sqrt{{(\frac{10}{2})}^{2} + {(10.8)}^{2}} + 10 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(10.8)}^{2}}$

Step 3.2

Divide $10$ by $2$ .

$50 + 5 \cdot \sqrt{5^{2} + {(10.8)}^{2}} + 10 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(10.8)}^{2}}$

Step 3.3

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

$50 + 5 \cdot \sqrt{25 + {(10.8)}^{2}} + 10 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(10.8)}^{2}}$

Step 3.4

Raise $10.8$ to the power of $2$ .

$50 + 5 \cdot \sqrt{25 + 116.64} + 10 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(10.8)}^{2}}$

Step 3.5

Add $25$ and $116.64$ .

$50 + 5 \cdot \sqrt{141.64} + 10 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(10.8)}^{2}}$

Step 3.6

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

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{5^{2}}{2^{2}} + {(10.8)}^{2}}$

Step 3.7

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

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{25}{2^{2}} + {(10.8)}^{2}}$

Step 3.8

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

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{25}{4} + {(10.8)}^{2}}$

Step 3.9

Raise $10.8$ to the power of $2$ .

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{25}{4} + 116.64}$

Step 3.10

To write $116.64$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{25}{4} + 116.64 \cdot \frac{4}{4}}$

Step 3.11

Combine $116.64$ and $\frac{4}{4}$ .

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{25}{4} + \frac{116.64 \cdot 4}{4}}$

Step 3.12

Combine the numerators over the common denominator.

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{25 + 116.64 \cdot 4}{4}}$

Step 3.13

Rewrite $\frac{25 + 116.64 \cdot 4}{4}$ in a factored form.

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Step 3.13.1

Multiply $116.64$ by $4$ .

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{25 + 466.56}{4}}$

Step 3.13.2

Add $25$ and $466.56$ .

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{\frac{491.56}{4}}$

Step 3.13.3

Divide $491.56$ by $4$ .

$50 + 5 \sqrt{141.64} + 10 \cdot \sqrt{122.89}$

$50 + 5 \sqrt{141.64} + 10 \sqrt{122.89}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$220.3621$