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A cylindrical glass has a volume of approximately 354 cm3354 \text{ cm}^3. The glass has a diameter of 6 cm6 \text{ cm} and is filled to 2 cm2\text{ cm} from the top with water. A golf ball 4 cm4\text{ cm} in diameter is placed into the glass. \newlineComplete the statements.\newline The space remaining in the glass is ____ and the volume of the golf ball is ____. \newlineTherefore, putting the golf ball in the glass of water ____ cause the water to overflow.

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Q. A cylindrical glass has a volume of approximately 354 cm3354 \text{ cm}^3. The glass has a diameter of 6 cm6 \text{ cm} and is filled to 2 cm2\text{ cm} from the top with water. A golf ball 4 cm4\text{ cm} in diameter is placed into the glass. \newlineComplete the statements.\newline The space remaining in the glass is ____ and the volume of the golf ball is ____. \newlineTherefore, putting the golf ball in the glass of water ____ cause the water to overflow.
  1. Calculate Glass Volume: Calculate the volume of the cylindrical glass. The formula for the volume of a cylinder is V=πr2hV = \pi r^2 h, where rr is the radius and hh is the height. The diameter of the glass is given as 66 cm, so the radius rr is half of that, which is 33 cm. The volume of the glass is given as 354354 cm3^3. We can use this information to find the height of the glass.
  2. Calculate Glass Height: Calculate the height of the glass using the volume.\newlineUsing the volume formula V=πr2hV = \pi r^2 h, we rearrange it to solve for hh: h=Vπr2h = \frac{V}{\pi r^2}. We plug in the values we have: h=354π⋅32h = \frac{354}{\pi \cdot 3^2}.\newlineh=354π⋅9h = \frac{354}{\pi \cdot 9}\newlineh=35428.274333882308138h = \frac{354}{28.274333882308138} (using π≈3.141592653589793\pi \approx 3.141592653589793)\newlineh≈12.52h \approx 12.52 cm
  3. Calculate Golf Ball Volume: Calculate the volume of the golf ball.\newlineThe formula for the volume of a sphere (which is the shape of a golf ball) is V=43πr3V = \frac{4}{3}\pi r^3. The diameter of the golf ball is given as 44 cm, so the radius rr is half of that, which is 22 cm. We plug in the values to find the volume of the golf ball.\newlineV=43π×23V = \frac{4}{3}\pi \times 2^3\newlineV=43π×8V = \frac{4}{3}\pi \times 8\newlineV≈43×3.141592653589793×8V \approx \frac{4}{3} \times 3.141592653589793 \times 8\newlineV≈33.510321638291124V \approx 33.510321638291124 cm3^3
  4. Calculate Remaining Space: Calculate the space remaining in the glass without the golf ball.\newlineSince the glass is filled to 2 cm2\,\text{cm} from the top with water, we subtract this from the total height of the glass to find the height of the water.\newlineHeight of water = Total height of glass - Space from top\newlineHeight of water = 12.52 cm−2 cm12.52\,\text{cm} - 2\,\text{cm}\newlineHeight of water = 10.52 cm10.52\,\text{cm}\newlineNow we calculate the volume of the water using the height of the water.\newlineVolume of water = πr2×\pi r^2 \times Height of water\newlineVolume of water = π×32×10.52\pi \times 3^2 \times 10.52\newlineVolume of water ≈3.141592653589793×9×10.52\approx 3.141592653589793 \times 9 \times 10.52\newlineVolume of water ≈295.305144033386 cm3\approx 295.305144033386\,\text{cm}^3
  5. Calculate Remaining Space After Golf Ball: Calculate the space remaining in the glass after the golf ball is placed in it.\newlineWe subtract the volume of the golf ball from the space remaining in the glass to see if there is enough space for the golf ball without causing the water to overflow.\newlineSpace remaining in glass = Volume of glass - Volume of water\newlineSpace remaining in glass = 354 cm3−295.305144033386 cm3354 \, \text{cm}^3 - 295.305144033386 \, \text{cm}^3\newlineSpace remaining in glass ≈58.694855966614 cm3\approx 58.694855966614 \, \text{cm}^3\newlineNow we compare this with the volume of the golf ball.\newlineSpace remaining after golf ball is placed = Space remaining in glass - Volume of golf ball\newlineSpace remaining after golf ball is placed ≈58.694855966614 cm3−33.510321638291124 cm3\approx 58.694855966614 \, \text{cm}^3 - 33.510321638291124 \, \text{cm}^3\newlineSpace remaining after golf ball is placed ≈25.184534328322876 cm3\approx 25.184534328322876 \, \text{cm}^3
  6. Determine Water Overflow: Determine if the water will overflow.\newlineIf the space remaining after the golf ball is placed is positive, the water will not overflow. If it is negative, the water will overflow.\newlineSince we have a positive value for the space remaining after the golf ball is placed, the water will not overflow.

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