Pre-Algebra Examples

 $\begin{array}{r} h = & 6 \\ l = & 3 \\ w = & 9 \end{array}$

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 $l = 3$ , the width $w = 9$ , and the height $h = 6$ into the formula for surface area of a pyramid.

$3 \cdot 9 + 9 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Multiply $3$ by $9$ .

$27 + 9 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.2

Apply the product rule to $\frac{3}{2}$ .

$27 + 9 \cdot \sqrt{\frac{3^{2}}{2^{2}} + {(6)}^{2}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.3

Raise $3$ to the power of $2$ .

$27 + 9 \cdot \sqrt{\frac{9}{2^{2}} + {(6)}^{2}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.4

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

$27 + 9 \cdot \sqrt{\frac{9}{4} + {(6)}^{2}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.5

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

$27 + 9 \cdot \sqrt{\frac{9}{4} + 36} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.6

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

$27 + 9 \cdot \sqrt{\frac{9}{4} + 36 \cdot \frac{4}{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.7

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

$27 + 9 \cdot \sqrt{\frac{9}{4} + \frac{36 \cdot 4}{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.8

Combine the numerators over the common denominator.

$27 + 9 \cdot \sqrt{\frac{9 + 36 \cdot 4}{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.9

Simplify the numerator.

Tap for more steps...

Step 3.9.1

Multiply $36$ by $4$ .

$27 + 9 \cdot \sqrt{\frac{9 + 144}{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.9.2

Add $9$ and $144$ .

$27 + 9 \cdot \sqrt{\frac{153}{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.10

Rewrite $\sqrt{\frac{153}{4}}$ as $\frac{\sqrt{153}}{\sqrt{4}}$ .

$27 + 9 \cdot \frac{\sqrt{153}}{\sqrt{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.11

Simplify the numerator.

Tap for more steps...

Step 3.11.1

Rewrite $153$ as $3^{2} \cdot 17$ .

Tap for more steps...

Step 3.11.1.1

Factor $9$ out of $153$ .

$27 + 9 \cdot \frac{\sqrt{9 (17)}}{\sqrt{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.11.1.2

Rewrite $9$ as $3^{2}$ .

$27 + 9 \cdot \frac{\sqrt{3^{2} \cdot 17}}{\sqrt{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.11.2

Pull terms out from under the radical.

$27 + 9 \cdot \frac{3 \sqrt{17}}{\sqrt{4}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.12

Simplify the denominator.

Tap for more steps...

Step 3.12.1

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

$27 + 9 \cdot \frac{3 \sqrt{17}}{\sqrt{2^{2}}} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.12.2

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

$27 + 9 \cdot \frac{3 \sqrt{17}}{2} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.13

Multiply $9 \frac{3 \sqrt{17}}{2}$ .

Tap for more steps...

Step 3.13.1

Combine $9$ and $\frac{3 \sqrt{17}}{2}$ .

$27 + \frac{9 (3 \sqrt{17})}{2} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.13.2

Multiply $3$ by $9$ .

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{{(\frac{9}{2})}^{2} + {(6)}^{2}}$

Step 3.14

Apply the product rule to $\frac{9}{2}$ .

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{9^{2}}{2^{2}} + {(6)}^{2}}$

Step 3.15

Raise $9$ to the power of $2$ .

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{81}{2^{2}} + {(6)}^{2}}$

Step 3.16

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

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{81}{4} + {(6)}^{2}}$

Step 3.17

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

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{81}{4} + 36}$

Step 3.18

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

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{81}{4} + 36 \cdot \frac{4}{4}}$

Step 3.19

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

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{81}{4} + \frac{36 \cdot 4}{4}}$

Step 3.20

Combine the numerators over the common denominator.

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{81 + 36 \cdot 4}{4}}$

Step 3.21

Simplify the numerator.

Tap for more steps...

Step 3.21.1

Multiply $36$ by $4$ .

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{81 + 144}{4}}$

Step 3.21.2

Add $81$ and $144$ .

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \sqrt{\frac{225}{4}}$

Step 3.22

Rewrite $\sqrt{\frac{225}{4}}$ as $\frac{\sqrt{225}}{\sqrt{4}}$ .

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \frac{\sqrt{225}}{\sqrt{4}}$

Step 3.23

Simplify the numerator.

Tap for more steps...

Step 3.23.1

Rewrite $225$ as $15^{2}$ .

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \frac{\sqrt{15^{2}}}{\sqrt{4}}$

Step 3.23.2

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

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \frac{15}{\sqrt{4}}$

Step 3.24

Simplify the denominator.

Tap for more steps...

Step 3.24.1

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

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \frac{15}{\sqrt{2^{2}}}$

Step 3.24.2

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

$27 + \frac{27 \sqrt{17}}{2} + 3 \cdot \frac{15}{2}$

Step 3.25

Multiply $3 (\frac{15}{2})$ .

Tap for more steps...

Step 3.25.1

Combine $3$ and $\frac{15}{2}$ .

$27 + \frac{27 \sqrt{17}}{2} + \frac{3 \cdot 15}{2}$

Step 3.25.2

Multiply $3$ by $15$ .

$27 + \frac{27 \sqrt{17}}{2} + \frac{45}{2}$

Step 4

To write $27$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$27 \cdot \frac{2}{2} + \frac{45}{2} + \frac{27 \sqrt{17}}{2}$

Step 5

Combine $27$ and $\frac{2}{2}$ .

$\frac{27 \cdot 2}{2} + \frac{45}{2} + \frac{27 \sqrt{17}}{2}$

Step 6

Combine the numerators over the common denominator.

$\frac{27 \cdot 2 + 45}{2} + \frac{27 \sqrt{17}}{2}$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $27$ by $2$ .

$\frac{54 + 45}{2} + \frac{27 \sqrt{17}}{2}$

Step 7.2

Add $54$ and $45$ .

$\frac{99}{2} + \frac{27 \sqrt{17}}{2}$

Step 8

Calculate the approximate solution to $4$ decimal places.

$105.1619$