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LaShay and Vincenzo walk into a dimly lit room within the Pyramidium House. They discover a secret chamber decorated with ancient hieroglyphs. At the center of the room awaits a mysterious pyramid emitting a faint glow. It seems that a riddle is inscribed on the walls. They take a closer look.

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LaShay and Vincenzo need to calculate the volume of the Guardian Pyramid to unlock the next part of their adventure. The pyramid is a regular triangular pyramid. The base has a side length of $5$ feet and a height of $4.33$ feet. The height of the pyramid is $12$ feet. Help them solve the mysterious riddle.

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After successfully solving the riddle of the Secret Chamber's Pyramid, LaShay and Vincenzo enter a room where the walls are constantly moving and reshaping the space. Amidst this changing environment, they notice several geometrically shaped vessels that are morphing in shape and size. One vessel, in the shape of a pentagonal pyramid, catches their attention.

The volume of the vessel, the side length of the base, and its apothem are known. However, the height of the vessel cannot be seen due to the shifting walls. LaShay and Vincenzo need to calculate the height of the vessel to stabilize the room. Help LaShay and Vincenzo solve it!

LaShay and Vincenzo have entered a room with a unique inverted square pyramid structure. The vertex of the pyramid is fixed to the ground, and its base extends upwards. This intriguing structure holds the key to their progress.

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Vincenzo and LaShay's eyes are widened as they enter the Pyramidium House's Enchanted Garden. A sense of mystery sends a tingle across their forearms. They are greeted by a maze of vibrant, magical plants and pyramidal trellises, each adorned with rare, sparkling flowers seemingly reaching for the skies. These mystical structures hold a crucial secret to their next challenge.

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Solution

Begin by considering the given diagram of the trellises.

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Each trellis is shaped like a pyramid and has a surface area of $117+9\sqrt{3}$ square meters. The base of the trellis is an equilateral triangle with sides of $6$ meters each. The surface area $SA$ of a pyramid can be found using the following formula.

$$\begin{array}{c}SA=\frac{1}{2}p\ell +B\end{array}$$

In this formula, $B$ is the base area, $p$ is the base perimeter, and $\ell$ is the slant height of the pyramid. The perimeter and area of the base must be found first to calculate the slant height of the trellis.

Perimeter of the Base

Since the base is an equilateral triangle with a side length of $6$ meters, multiply this side length by $3$ to get the perimeter of the base.

$$\begin{array}{c}\text{Perimeter of the Base}\\ 6\cdot 3=18\text{ meters}\end{array}$$

Area of the Base

The area of a triangle is calculated as half the product of its base and height.

$$\begin{array}{c}A=\frac{1}{2}bh\end{array}$$

This means that the height of the triangle must be determined first. The height of an equilateral triangle is a perpendicular segment that extends from one vertex to the opposite side, effectively bisecting that side.

When drawing the height of this triangle, two right triangles are formed. Each of these right triangles has a base of $3$ meters, which is half the length of the equilateral triangle's side. Substituting $c=6$ and $b=3$ into the Pythagorean Theorem, the missing side $a$ of these right triangles can be determined.

$c^2=a^2+b^2$

SubstituteII

$c=6$, $b=3$

$6^2=a^2+3^2$

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Solve for $a$

CalcPow

Calculate power

$36=a^2+9$

SubEqn

$\text{LHS}-9=\text{RHS}-9$

$27=a^2$

RearrangeEqn

Rearrange equation

$a^2=27$

SqrtEqn

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

$a=\pm \sqrt{27}$

SplitIntoFactors

Split into factors

$a=\pm \sqrt{9\cdot 3}$

SqrtProd

$\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}$

$a=\pm \sqrt{9}\cdot \sqrt{3}$

CalcRoot

Calculate root

$a=\pm 3\sqrt{3}$

$a>0$

$a=3\sqrt{3}$

The value of $a$ corresponds to the height of the equilateral triangle. Then, height $h$ of the equilateral triangle is $3\sqrt{3}$ meters. Next, substitute the base and height of the equilateral triangle into the formula for the area of a triangle to determine its area.

$A=\frac{1}{2}b\cdot h$

SubstituteII

$b=6$, $h=3\sqrt{3}$

$A=\frac{1}{2}\cdot 6\cdot 3\sqrt{3}$

Multiply

Multiply

$A=\frac{1}{2}\cdot 18\sqrt{3}$

MoveRightFacToNumOne

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

$A=\frac{18\sqrt{3}}{2}$

CalcQuot

Calculate quotient

$A=9\sqrt{3}$

The area of this triangle is $9\sqrt{3}$ square meters, which corresponds to the area of the base $B$ of the pyramid.

Calculating the Slant Height

The base area, the base perimeter, and surface area can now be substituted into the formula for the surface area of a pyramid. Then, the obtained equation can be solved for $\ell$ to find the slant height of the trellises.

$SA=\frac{1}{2}p\cdot \ell +B$

SubstituteValues

Substitute values

$117+9\sqrt{3}=\frac{1}{2}\cdot 18\cdot \ell +9\sqrt{3}$

SubEqn

$\text{LHS}-9\sqrt{3}=\text{RHS}-9\sqrt{3}$

$117=\frac{1}{2}\cdot 18\ell$

MoveRightFacToNumOne

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

$117=\frac{18\ell }{2}$

CalcQuot

Calculate quotient

$117=9\ell$

RearrangeEqn

Rearrange equation

$9\ell =117$

DivEqn

$\text{LHS}/9=\text{RHS}/9$

$\ell =13$

The slant height measures $13$ meters. This indicates that LaShay and Vincenzo must choose door number $13$ to access their final challenge — the Celestial Observatory.

LaShay and Vincenzo finally reach the last passage of the Pyramidium House — the Celestial Observatory. They are awe-stricken by a pyramid-shaped map that displays the constellations in a three-dimensional layout. To activate the map's prophecy, they must find the volume of the map.

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After completing the last adventure of the Pyramidium House, LaShay and Vincenzo enter the heart of the structure. They recall their initial challenge of needing to pour a potion from a prism container into two smaller pyramid containers.

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Using their newfound knowledge and the given dimensions, they can calculate whether the portion in the prism is enough to fill both pyramids.