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A cylindrical glass has a volume of approximately 502 cm3502 \text{ cm}^3. The glass has a diameter of 8 cm8 \text{ cm} and is filled to 1cm1\text{cm} from the top with water. A golf ball 4cm4\text{cm} in diameter is placed into the glass. Complete the statements. The space remaining in the glass is [DROP DOWN 11] and the volume of the golf ball is [DROP DOWN 22]. Therefore, putting the golf ball in the glass of water [DROP DOWN 33] cause the water to overflow.

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Q. A cylindrical glass has a volume of approximately 502 cm3502 \text{ cm}^3. The glass has a diameter of 8 cm8 \text{ cm} and is filled to 1cm1\text{cm} from the top with water. A golf ball 4cm4\text{cm} in diameter is placed into the glass. Complete the statements. The space remaining in the glass is [DROP DOWN 11] and the volume of the golf ball is [DROP DOWN 22]. Therefore, putting the golf ball in the glass of water [DROP DOWN 33] cause the water to overflow.
  1. Calculate Water Volume: First, we need to calculate the volume of the cylindrical glass that is actually filled with water. Since the glass is filled to 1 cm1 \, \text{cm} from the top, we need to find the height of the water in the glass.\newlineWe know the total volume of the glass and the diameter, so we can use the formula for the volume of a cylinder: V=πr2hV = \pi r^2 h, where VV is the volume, rr is the radius, and hh is the height.\newlineThe radius of the glass is half of the diameter, so r=8 cm/2=4 cmr = 8 \, \text{cm} / 2 = 4 \, \text{cm}.\newlineWe can rearrange the formula to solve for the height: h=V/(πr2)h = V / (\pi r^2).
  2. Calculate Water Height: Now we calculate the height of the water in the glass using the volume of 502 cm3502 \, \text{cm}^3. \newlineh=502 cm3π×(4 cm)2h = \frac{502 \, \text{cm}^3}{\pi \times (4 \, \text{cm})^2}\newlineh=502 cm3π×16 cm2h = \frac{502 \, \text{cm}^3}{\pi \times 16 \, \text{cm}^2}\newlineh=502 cm350.265 cm2 (using π≈3.14159)h = \frac{502 \, \text{cm}^3}{50.265 \, \text{cm}^2} \, (\text{using} \, \pi \approx 3.14159)\newlineh≈9.98 cmh \approx 9.98 \, \text{cm}\newlineSince the glass is filled to 1 cm1 \, \text{cm} from the top, this calculation confirms that the height of the water is approximately correct.
  3. Calculate Golf Ball Volume: Next, we calculate the volume of the golf ball using the formula for the volume of a sphere: V=43πr3V = \frac{4}{3}\pi r^3, where rr is the radius of the golf ball.\newlineThe radius of the golf ball is half of its diameter, so r=4 cm2=2 cmr = \frac{4 \text{ cm}}{2} = 2 \text{ cm}.\newlineV=43π×(2 cm)3V = \frac{4}{3}\pi \times (2 \text{ cm})^3\newlineV=43π×8 cm3V = \frac{4}{3}\pi \times 8 \text{ cm}^3\newlineV=43×3.14159×8 cm3V = \frac{4}{3} \times 3.14159 \times 8 \text{ cm}^3\newlineV≈33.51 cm3V \approx 33.51 \text{ cm}^3
  4. Check for Overflow: Now we need to determine if placing the golf ball in the glass will cause the water to overflow.\newlineThe volume of water displaced by the golf ball will be equal to the volume of the golf ball.\newlineSince the volume of the golf ball is 33.51 cm333.51\,\text{cm}^3, and the glass was filled to 1 cm1\,\text{cm} from the top, we need to check if this additional volume will exceed the space left in the glass.\newlineThe total volume of the glass minus the volume filled with water gives us the space left: 502 cm3−(502 cm3−33.51 cm3)=33.51 cm3502\,\text{cm}^3 - (502\,\text{cm}^3 - 33.51\,\text{cm}^3) = 33.51\,\text{cm}^3.
  5. Final Conclusion: Since the space left in the glass is exactly equal to the volume of the golf ball, placing the golf ball in the glass will not cause the water to overflow.

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