Sanjay makes souvenir pyramids by pouring liquid into a pyramid-shaped mold. The mold he uses has a square base with a side length of 10\text{ cm}10 cm10, start text, space, c, m, end text, and the height of the mold is 10\text{ cm}10 cm10, start text, space, c, m, end text. Sanjay wants to make a smaller pyramid using the same mold, so he plans to fill the mold 2\text{ cm}2 cm2, start text, space, c, m, end text from the top. What is the approximate volume of this smaller pyramid?

If we fill the pyramid mold 2\text{ cm}2 cm2, start text, space, c, m, end text from the top, we have a smaller pyramid that's similar to the original pyramid.

Since the pyramids are similar, we can set up a proportional equation to find the side lengths and height of the smaller pyramid, and then find its volume.

Hint #22 / 4

Base and height of smaller pyramid

The height of the smaller pyramid is 10-2=8\text{ cm}10−2=8 cm10, minus, 2, equals, 8, start text, space, c, m, end text.

We can solve for the length \blueE{\ell}ℓstart color #0c7f99, ell, end color #0c7f99 in the smaller pyramid using a proportional equation.

\begin{aligned} \dfrac{\blueE{\ell}}{10} &= \dfrac{8}{10} \\\\ \blueE{\ell} &= \blueE{8} \end{aligned}

10

ℓ

=

10

8

=8

Hint #33 / 4

Volume of smaller pyramid

\begin{aligned} \text{volume}_{\text{pyramid}} &= \dfrac13(\text{base area})(\text{height}) \\\\ &= \dfrac13 \cdot (\blueE{\ell})^2\cdot (\text{height}) \\\\ &= \dfrac13 \cdot \blueE{8}^2\cdot(8)\\\\ &= \dfrac{512}{3}=170.\overline{6}\\\\ &\approx \purpleD{170.67} \end{aligned}

volume

pyramid

=

3

1

(base area)(height)

=

3

1

⋅(ℓ)

2

⋅(height)

=

3

1

⋅8

2

⋅(8)

=

3

512

=170.

6

≈170.67

Hint #44 / 4

To the nearest cubic centimeter, the volume of the smaller pyramid is about 171\text{ cm}^3171 cm

3

171, start text, space, c, m, end text, cubed.