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Tadeo is getting ready to go camping. He has a pyramid-shaped tent with a regular pentagonal base.

Solution

Start by finding the area of the base of the tent. Then, the formula for the volume of a pyramid will be used.

Area of the Base

The base of the tent is a regular pentagon with side lengths $1.4$ meters. Therefore, its perimeter $p$ is $5$ times $1.4.$

$$\begin{array}{c}p=5\times 1.4=7\text{meters}\end{array}$$

Now, draw the apothem of the pentagonal base. The measure of each central angle of a regular pentagon is $72^{\circ }$ and is bisected by the apothem.

The apothem is perpendicular to any side of the polygon and bisects it. As a result, a right triangle with a leg of $0.7$ meters is formed.

The value of $a$ can be written in terms of the tangent ratio of $36^{\circ }.$

$$\begin{array}{r}\tan 36^{\circ }=\frac{0.7}{a}\iff a=\frac{0.7}{\tan 36^{\circ }}\end{array}$$

Finally, to find the area of the base $B,$ substitute $a=\frac{0.7}{\tan 36^{\circ }}$ and $p=7$ into the area formula for regular polygons.

$B=\frac{1}{2}ap$

SubstituteII

$a=\frac{0.7}{\tan 36^{\circ }}$, $p=7$

$B=\frac{1}{2}\left(\frac{0.7}{\tan 36^{\circ }}\right)(7)$

Evaluate right-hand side

MoveRightFacToNum

$\frac{a}{c}\cdot b=\frac{a\cdot b}{c}$

$B=\frac{1}{2}\left(\frac{4.9}{\tan 36^{\circ }}\right)$

MultFrac

Multiply fractions

$B=\frac{4.9}{2\tan 36^{\circ }}$

UseCalc

Use a calculator

$B=3.372135\dots$

RoundDec

Round to $2$ decimal place(s)

$B=3.37$

The area of the base is about $3.37$ square meters.

Volume of the Tent

Now that the base area and height are known, substitute these values into the formula for the volume of a pyramid.

$V=\frac{1}{3}Bh$

SubstituteII

$B=3.37$, $h=1.6$

$V=\frac{1}{3}(3.37)(1.6)$

Evaluate right-hand side

Multiply

Multiply

$V=\frac{1}{3}\cdot 5.392$

MoveRightFacToNumOne

$\frac{1}{b}\cdot a=\frac{a}{b}$

$V=\frac{5.392}{3}$

CalcQuot

Calculate quotient

$V=1.797333\dots$

RoundDec

Round to $2$ decimal place(s)

$V\approx 1.80$

The volume of the tent is about $1.80$ cubic meters.

Architects might enjoy turning things upside down. Maya is interested in architecture and follows some online magazines about it. After reading an article about the Slovak Radio Building in Bratislava, Slovakia, she wonders about the area of the square rooftop.

External credits: Thomas Ledl

If the height of the building is $80$ meters and its volume is about $245760$ cubic meters, find the area of the rooftop of the building.

Designers and inventors also benefit from pyramids. An object attracts Mark's attention on a school trip to a maritime museum. The guide explains that it is called a deck prism, which was invented to illuminate the cabins below deck before electric lighting. Mark buys a replica of the deck prism, which is composed of a prism and pyramid, each with a regular hexagonal base.

Maya's father decides to cover the roof of their house with waterproof insulation material. Maya's father asks Maya to calculate how many square feet of insulation material is needed.

The roof is a square pyramid with a height of $8$ feet and base side length of $30$ feet. Help Maya calculate the area.

Solution

Maya needs to calculate the lateral area of the square pyramid. To do so, she first needs to calculate the slant height $\ell$ of the pyramid, which can be found by using the Pythagorean Theorem.

The height $h$ of the pyramid is the distance between the vertex and the center of the base, and $b$ is half the base side length. Therefore, $b=\frac{30}{2}=15$ feet.

${\ell }^2=b^2+h^2$

SubstituteII

$b=15$, $h=8$

${\ell }^2={15}^2+8^2$

Solve for $\ell$

CalcPow

Calculate power

${\ell }^2=225+64$

AddTerms

Add terms

${\ell }^2=289$

SqrtEqn

$\sqrt{\text{LHS}}=\sqrt{\text{RHS}}$

$\ell =\sqrt{289}$

CalcRoot

Calculate root

$\ell =17$

Since a negative value does not make sense in this context, only the principal root is considered. Therefore, the slant height is $17$ feet. Next, the perimeter of the base will be found. Since the base is a square, its perimeter $p$ is $4$ times the base side length.

$$\begin{array}{c}p=4\cdot 30=120\end{array}$$

Finally, the lateral area of the pyramid can be found by substituting $p=120$ and $\ell =17$ into the formula.

$LA=\frac{1}{2}p\ell$

SubstituteII

$p=120$, $\ell =17$

$LA=\frac{1}{2}(120)(17)$

Evaluate right-hand side

Multiply

Multiply

$LA=\frac{1}{2}\cdot 2040$

MoveRightFacToNumOne

$\frac{1}{b}\cdot a=\frac{a}{b}$

$LA=\frac{2040}{2}$

CalcQuot

Calculate quotient

$LA=1020$

The amount of material needed to cover the roof is $1020$ square feet.