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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 10cm, and the height of the mold is 10cm. Sanjay wants to make a smaller pyramid using the same mold, so he plans to fill the mold 2cm from the top. What is the approximate volume of this smaller pyramid? Round to the nearest cubic centimeter.

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 cm 10 \mathrm{~cm} , and the height of the mold is 10 cm 10 \mathrm{~cm} .\newlineSanjay wants to make a smaller pyramid using the same mold, so he plans to fill the mold 2 cm 2 \mathrm{~cm} from the top.\newlineWhat is the approximate volume of this smaller pyramid?\newlineRound to the nearest cubic centimeter.

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Full solution

Q. 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 cm 10 \mathrm{~cm} , and the height of the mold is 10 cm 10 \mathrm{~cm} .\newlineSanjay wants to make a smaller pyramid using the same mold, so he plans to fill the mold 2 cm 2 \mathrm{~cm} from the top.\newlineWhat is the approximate volume of this smaller pyramid?\newlineRound to the nearest cubic centimeter.
  1. Understand Problem & Formula: Understand the problem and the formula for the volume of a pyramid. The volume of a pyramid is given by the formula V=(13)×base area×heightV = (\frac{1}{3}) \times \text{base area} \times \text{height}. Since the base is square, the base area is the side length squared.
  2. Calculate Full-Size Pyramid: Calculate the volume of the full-size pyramid using the given dimensions.\newlineThe side length of the base ss is 10 cm10\,\text{cm}, and the height hh is also 10 cm10\,\text{cm}.\newlineUsing the formula V=13×s2×hV = \frac{1}{3} \times s^2 \times h, we get V=13×10 cm×10 cm×10 cmV = \frac{1}{3} \times 10\,\text{cm} \times 10\,\text{cm} \times 10\,\text{cm}.
  3. Perform Calculation: Perform the calculation for the full-size pyramid. V=13×10 cm×10 cm×10 cm=13×1000 cm3=333.33 cm3V = \frac{1}{3} \times 10\,\text{cm} \times 10\,\text{cm} \times 10\,\text{cm} = \frac{1}{3} \times 1000\,\text{cm}^3 = 333.33\,\text{cm}^3 (rounded to two decimal places).
  4. Determine Scale Factor: Determine the scale factor for the smaller pyramid.\newlineSince Sanjay is filling the mold 2 cm2\,\text{cm} from the top, the height of the smaller pyramid is 10 cm−2 cm=8 cm10\,\text{cm} - 2\,\text{cm} = 8\,\text{cm}. The scale factor for the height is 8 cm10 cm=0.8\frac{8\,\text{cm}}{10\,\text{cm}} = 0.8.
  5. Apply Scale Factor: Apply the scale factor to the base dimensions.\newlineThe scale factor for the volume is the cube of the scale factor for the height because volume is a three-dimensional measure. Therefore, the scale factor for the volume is 0.830.8^3.
  6. Calculate Scale Factor: Calculate the scale factor for the volume. 0.83=0.5120.8^3 = 0.512.
  7. Apply to Find Smaller Pyramid Volume: Apply the scale factor to the volume of the full-size pyramid to find the volume of the smaller pyramid.\newlineThe volume of the smaller pyramid is 0.512×333.33 cm30.512 \times 333.33\,\text{cm}^3.
  8. Perform Calculation for Smaller Pyramid: Perform the calculation for the volume of the smaller pyramid. Vsmall=0.512×333.33 cm3≈170.67 cm3V_{\text{small}} = 0.512 \times 333.33\,\text{cm}^3 \approx 170.67\,\text{cm}^3.
  9. Round Smaller Pyramid Volume: Round the volume of the smaller pyramid to the nearest cubic centimeter. Vsmall≈171 cm3V_{\text{small}} \approx 171\,\text{cm}^3 when rounded to the nearest cubic centimeter.

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